someone drew a triangle and three segments across it. Each segment started at a vertex and stopped at the midpoint of the opposite side. The segments met in a point. The person was impressed and repeated the experiment on a different shape of triangle. Again the segments met in a point. The person drew yet a third triangle, very carefully, with the same result. He told his friends. To their surprise and delight, the coincidence worked for them, too. Word spread, and the magic of the three segments was regarded as the work of a higher power.

Centuries passed, and someone proved that the three medians do indeed concur in a point, now called the centroid. The ancients found other points, too, now called the incenter, circumcenter, and orthocenter. More centuries passed, more special points were discovered, and a definition of triangle center emerged. Like the definition of continuous function, this definition is satisfied by infinitely many objects, of which only finitely many will ever be published. The Encyclopedia of Triangle Centers (ETC) extends a list of 400 triangle centers published in the 1998 book Triangle Centers and Central Triangles. For subsequent developments, click Links (one of the buttons atop this page). In particular, Eric Weisstein's MathWorld, covers much of classical and modern triangle geometry, including sketches and references.

A site in which triangle centers play a central role is Bernard Gibert's Cubics in the Triangle Plane. Special points and properties of 4-sided plane figures are closely associated with triangle centers; see Chris van Tienhoven's Encyclopedia of Quadri-Figures (EQF).





HOW TO USE ETC

You won't have to scroll down very far to find well known centers. Other named centers can be found using your computer's searcher - for example, search for "Apollonius" to find "Apollonius point" as X(181).

To determine if a possibly new center is already listed, click Tables at the top of this page and scroll to "Search 6.9.13". If you're unsure of a term, click Glossary or Pierre Douillet's much expanded and very useful version: Translation of the Kimberling's Glossary into barycentrics.

For visual constructions of selected centers with text, click Sketches. To learn about the triangle geometry interest group, Hyacinthos and other resources, or to view acknowledgments or supplementary encyclopedic material, click Links, Thanks, or Tables.

Under, you can find(and two other Searches), which can be used to determine whether a newly discovered point is already in ETC. For such a search, be sure to visit Ron Knott's Triangle Convertor for Cartesian, Trilinear and Barycentric Coordinates

If you have The Geometer's Sketchpad, you can view sketches of many of the triangle centers. These are dynamic sketches, meaning that you can vary the shape of the reference triangle A, B, C by dragging these vertices. (For information on Sketchpad, click Sketchpad.) The sketches are also useful for making your own Sketchpad tools, so that you can quickly construct X-of-T for many choices of X and T. For example, starting with ABC and point P, you could efficiently construct center X of the four triangles ABC, BCP, CAP, ABP.

The algebraic definition of triangle center (MathWorld) admits points whose geometric interpretation for fixed numerical sidelengths a,b,c is not "central." Roger Smyth offers this example: on the domain of scalene triangles, define f(a,b,c) = 1 for a>b and a>c and f(a,b,c) = 0 otherwise; then f(a,b,c) : f(b,c,a) : f(c,a,b) is a triangle center which picks out the vertex opposite the longest side. Such centers turn out to be useful, as, for example, when distinguishing between the Fermat point and the 1st isogonic center; see the note at X(13).

NOTATION AND COORDINATES

The reference triangle is ABC, with sidelengths a,b,c. Each triangle center P has homogeneous trilinear coordinates, or simply trilinears, of the form x : y : z. This means that there is a nonzero function h of (a,b,c) such that

x = hx', y = hy', z = hz',



where x', y', z' denote the directed distances from P to sidelines BC, CA, AB. Likewise, u : v : w are barycentrics if there is a function k of (a,b,c) such that

u = ku', v = kv', w = kw',



where u', v', w' denote the signed areas of triangles PBC, PCA, PAB. Both coordinate systems are widely used; if trilinears for a point are x : y : z, then barycentrics are ax : by : cz.

In order that every center should have its own name, in cases where no particular name arises from geometrical or historical considerations, the name of a star is used instead. For example, X(770) is POINT ACAMAR. For a list of star names, visit SkyEye - (Un)Common Star Names.

Introduced on December 23, 2019: Alphabetical Index of Terms in ETC

Alphabetical Index of Terms in ETC, by César Lozada

Introduced on May 2, 2019: Writer

If you wish to submit one or more triangles centers for possible inclusion in ETC, please click Tables at the top of this page, then scroll to and click Search_13_6_9. There, find Writer, to be used for proper formatting.

Introduced on December 28, 2016: Index of Triangles Referenced in ETC

Many triangles are defined in the plane of a reference triangle ABC. Some of them have well-established names (e.g., medial, orthic, tangential), but many more have been discovered only recently.

The Index is authored and updated by César Lozada. You can access it here, and also from Glossary and Tables.

Introduced on March 21, 2015: Shinagawa coefficients for triangle centers on the Euler line

Suppose that X is a triangle center given by barycentric coordinates f(a,b,c) : f(b,c,a) : f(c,a,b). The Shinagawa coefficients of X are the functions G(a,b,c) and H(a,b,c) such that

f(a,b,c) = G(a,b,c)*S2 + H(a,b,c)*S B S C .

For many choices of X, G(a,b,c) and H(a,b,c) are conveniently expressed in terms of the following:

E = (S B + S C )(S C + S A )(S A + S B )/S2, so that E = (abc/S)2 = 4R2

F = S A S B S C /S2, so that F = (a2 + b2 + c2)/2 - 4R2 = S ω - 4R2

Examples:

X(2) has Shinagawa coefficients (1, 0); i.e., X(2) = 1*S2 + 0*S B S C

X(3) has Shinagawa coefficients (1, -1)

X(4) has Shinagawa coefficients (0, 1)

X(5) has Shinagawa coefficients (1, 1)

X(23) has Shinagawa coefficients (E + 4F, -4E - 4F)

X(1113) has Shinagawa coefficients (R - |OH|, -3R + |OH|)



A cyclic sum notation, $...$, is introduced here especially for use with Shinagawa coefficients. For example, $aS B S C $ abbreviates aS B S C + bS C S A + cS A S B .

Example: X(21) has Shinagawa coefficients ($aS A $, abc - $aS A $)

If a point X has Shinagawa coefficients (u,v) where u and v are real numbers (i.e, G(a,b,c) and H(a,b,c) are constants), then the segment joining X and X(2) is given by |GX| = 2v|GO|/(3u + v), where |GO| = (E - 8F)1/2/6. Then the equation |GX| = 2v|GO|/(3u + v) can be used to obtain these combos:

X = [(u + v)/2]*X(2) - (v/3)*X(3)

X = u*X(2) + (v/3)*X(4)

X = u*X(3) + [(u + v)/2]*X(4).

The function F is also given by these identities:

F = (4R2 - 36|GO|2)/8 and F = R2( 1 - J2)/2, where J = |OH|/R.

Introduced on November 1, 2011: Combos

Suppose that P and U are finite points having normalized barycentric coordinates (p,q,r) and (u,v,w). (Normalized means that p + q + r = 1 and u + v + w = 1.) Suppose that f = f(a,b,c) and g = g(a,b,c) are nonzero homogeneous functions having the same degree of homogeneity. Let x = fp + gu, y = fq + gv, z = fr + gw. The (f,g) combo of P and U, denoted by f*P + g*U, is introduced here as the point X = x : y : z (homogeneous barycentric coordinates); the normalized barycentric coordinates of X are (kx,ky,kz), where k=1/(x+y+z).

Note 1. If P and U are given by normalized trilinear coordinates (instead of barycentric), then f*P + g*U has homogeneous trilinears fp+gu : fq+gv : fr+gw, which is symbolically identical to the homogenous barycentrics for f*P + g*U. The normalized trilinear coordinates for X are (hx,hy,hz), where h=2*area(ABC)/(ax + by + cz).

Note 2. The definition of combo readily extends to finite sets of finite points. In particular, the (f,g,h) combo of P = (p,q,r), U = (u,v,w), J = (j,k,m) is given by fp + gu + hj : fq + gv + hk : fr + gw + hm and denoted by f*P + g*U + h*J.

Note 3. f*P + g*U is collinear with P and U, and its {P,Q}-harmonic conjugate is fp - gu : fq - gv : fr - gw.

Note 4. Suppose that f,g,h are homogeneous symmetric functions all of the same degree of homogeneity, and suppose that X, X', X" are triangle centers. Then f*X + g*X' + h*X'' is a triangle center.

Note 5. Suppose that X, X', X'', X''' are triangle centers and X', X'', X''' are not collinear. Then there exist f,g,h as in Note 4 such that X = f*X' + g*X'' + h*X'''. That is, loosely speaking, every triangle center is a linear combination of any other three noncollinear triangle centers.

Note 6. Continuing from Note 5, examples of f,g,h are conveniently given using Conway symbols for a triangle ABC with sidelengths a,b,c. Conway symbols and certain classical symbols are identified here:

S = 2*area(ABC)

S A = (b2 + c2 - a2)/2 = bc cos A

S B = (c2 + a2 - b2)/2 = ca cos B

S C = (a2 + b2 - c2)/2 = ab cos C

S ω = S cot ω

s = (a + b + c)/2

s a = (b + c - a)/2

s b = (c + a - b)/2

s c = (a + b - c)/2

r = inradius = S/(a + b + c)

R = circumradius = abc/(2S)

cot(ω) = (a2 + b2 + c2)/(2S), where ω is the Brocard angle

Note 7. The definition of combo along with many examples were developed by Peter Moses prior to November 1, 2011. After that combos have been further developed by Peter Moses, Randy Hutson, and Clark Kimberling.

Examples of two-point combos:

X(175) = 2s*X(1) - (r + 4R)*X(7)

X(176) = 2s*X(1) + (r + 4R)*X(7)

X(481) = s*X(1) - (r + 4R)*X(7)

X(482) = s*X(1) + (r + 4R)*X(7)

Examples of three-point combos: see below at X(1), X(2), etc.

Note 8. Suppose that T is a (central) triangle with vertices A',B',C' given by normalized barycentrics. Then T is represented by a 3x3 matrix with row sums equal to 1. Let NT denote the set of these matrices and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion, so that (NT, *) is a group. Once normalized, any central T can be used to produce triangle centers as combos of the form Xcom(nT); see the preambles to X(3663) and X(3739).

X(1) = INCENTER

Trilinears 1 : 1 : 1Barycentrics a : b : cBarycentrics sin A : sin B : sin CTripolars Sqrt[b c (b + c - a)] : :Tripolars sec A' : :, where A'B'C' is the excentral triangle

X(1) is the point of concurrence of the interior angle bisectors of ABC; the point inside ABC whose distances from sidelines BC, CA, AB are equal. This equal distance, r, is the radius of the incircle, given by r = 2*area(ABC)/(a + b + c).

Three more points are also equidistant from the sidelines; they are given by these names and trilinears:

A-excenter = -1 : 1 : 1, B-excenter = 1 : -1 : 1, C-excenter = 1 : 1 : -1.

The radii of the excircles are 2*area(ABC)/(-a + b + c), 2*area(ABC)/(a - b + c), 2*area(ABC)/(a + b - c).

If you have The Geometer's Sketchpad, you can view Incenter.

If you have GeoGebra, you can view Incenter.

Writing the A-exradius as r(a,b,c), the others are r(b,c,a) and r(c,a,b). If these exradii are abbreviated as r a , r b , r c , then 1/r = 1/r a +1/r b + 1/r c . Moreover,

area(ABC) = sqrt(r*r a *r b *r c ) and r a + r b + r c = r + 4R, where R denotes the radius of the circumcircle.

The incenter is the identity of the group of triangle centers under trilinear multiplication defined by (x : y : z)*(u : v : w) = xu : yv : zw.

A construction for * is easily obtained from the construction for "barycentric multiplication" mentioned in connection with X(2), just below.

The incenter and the other classical centers are discussed in these highly recommended books:

Paul Yiu, Introduction to the Geometry of the Triangle, 2002;

Nathan Altshiller Court, College Geometry, Barnes & Noble, 1952;

Roger A. Johnson, Advanced Euclidean Geometry, Dover, 1960.

Let O A be the circle tangent to side BC at its midpoint and to the circumcircle on the side of BC opposite A. Define O B and O C cyclically. Let L A be the external tangent to circles O B and O C that is nearest to O A . Define L B and L C cyclically. Let A' = L B ∩L C , and define B' and C' cyclically. Then triangle A'B'C' is homothetic to the medial triangle, and the center of homothety is X(1); see the reference at X(1001).

Let A'B'C' and A"B"C" be the intouch and excentral triangles; X(1) is the radical center of the circumcircles of AA'A", BB'B", CC'C". (Randy Hutson, December 10, 2016)

Let A'B'C' be the Feuerbach triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let A" be the trilinear pole of line B'C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1). (Randy Hutson, November 17, 2019)

Let A'B'C' be the mixtilinear excentral triangle. Let L A be the trilinear polar of A', and define L B and L C cyclically. Let A" = L B ∩L C , and define B" and C" cyclically. The lines AA", BB", CC" concur in X(1). (Randy Hutson, November 17, 2019)

Let O A be the circle centered at the A-vertex of the excenters-midpoints triangle and passing through A; define O B and O C cyclically. X(1) is the radical center of O A , O B , O C . (Randy Hutson, August 30, 2020)

Let O A be the circle centered at the A-vertex of the Gemini triangle 22 and passing through A; define O B and O C cyclically. X(1) is the radical center of O A , O B , O C . (Randy Hutson, August 30, 2020)

X(1) lies on all Z-cubics (e.g., Thomson, Darboux, Napoleon, Neuberg) and these lines: 2,8 3,35 4,33 5,11 6,9 7,20 15,1251 16,1250 19,28 21,31 24,1061 25,1036 29,92 30,79 32,172 39,291 41,101 49,215 54,3460 60,110 61,203 62,202 64,1439 69,1245 71,579 74,3464 75,86 76,350 82,560 84,221 87,192 88,100 90,155 99,741 102,108 104,109 142,277 147,150 159,1486 163,293 164,258 166,1488 167,174 168,173 179,1142 181,970 182,983 184,1726 185,296 188,361 190,537 195,3467 196,207 201,212 204,1712 210,2334 224,377 227,1465 228,1730 229,267 256,511 257,385 280,1256 281,282 289,363 312,1089 318,1897 320,752 321,964 329,452 335,384 336,811 341,1050 344,1265 346,1219 357,1508 358,1507 364,365 371,1702 372,1703 376,553 378,1063 393,836 394,1711 399,3065 409,1247 410,1248 411,1254 442,1834 474,1339 475,1861 512,875 513,764 514,663 522,1459 528,1086 561,718 563,1820 564,1048 572,604 573,941 574,1571 594,1224 607,949 615,3300 631,1000 644,1280 647,1021 650,1643 651,1156 659,891 662,897 672,1002 689,719 704,1502 727,932 731,789 748,756 761,825 765,1052 810,1577 840,1308 905,1734 908,998 921,1800 939,1260 945,1875 947,1753 951,1435 969,1444 971,1419 989,1397 1013,1430 1037,1041 1053,1110 1057,1598 1059,1597 1073,3341 1075,1148 1106,1476 1157,3483 1168,1318 1170,1253 1185,1206 1197,1613 1292,1477 1333,1761 1342,1700 1343,1701 1361,1364 1389,1393 1399,1727 1406,1480 1409,1765 1437,1710 1472,1791 1719,1790 1855,1886 1859,1871 1872,1887 2120,3461 2130,3347 3183,3345 3342,3343 3344,3351 3346,3353 3348,3472 3350,3352 3354,3355 3462,3469

X(1) is the {X(2),X(8)}-harmonic conjugate of X(10). For a list of other harmonic conjugates of X(1), click Tables at the top of this page.

X(1) = midpoint of X(i) and X(j) for these (i,j): (3, 1482), (7,390), (8,145), (55,2099), (56,2098)

X(1) = reflection of X(i) in X(j) for these (i,j): (2,551), (3,1385), (4,946), (6,1386), (8,10), (9,1001), (10,1125), (11,1387), (36,1319), (40,3), (43,995), (46,56), (57,999), (63,993), (65,942), (72,960), (80,11), (100,214), (191,21), (200,997), (238,1279), (267,229), (291,1015), (355,5), (484,36), (984,37), (1046,58), (1054,106), (1478,226)

X(1) = isogonal conjugate of X(1)

X(1) = isotomic conjugate of X(75)

X(1) = cyclocevian conjugate of X(1029)

X(1) = circumcircle-inverse of X(36)

X(1) = Fuhrmann-circle-inverse of X(80)

X(1) = Bevan-circle-inverse of X(484)

X(1) = Spieker-radical-circle-inverse of X(38471)

X(1) = complement of X(8)

X(1) = anticomplement of X(10)

X(1) = anticomplementary conjugate of X(1330)

X(1) = complementary conjugate at X(1329)

X(1) = eigencenter of cevian triangle of X(i) for I = 1, 88, 100, 162, 190

X(1) = eigencenter of anticevian triangle of X(i) for I = 1, 44, 513

X(1) = exsimilicenter of inner and outer Soddy circles; insimilicenter is X(7)

X(1) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,9), (4,46), (6,43), (7,57), (8,40), (9,165), (10,191), (21,3), (29,4), (75,63), (77,223), (78,1490), (80,484), (81,6), (82,31), (85,169), (86,2), (88,44), (92,19), (100,513), (104,36), (105,238), (174,173), (188,164), (220,170), (259,503), (266,361), (280,84), (366,364), (508,362), (1492,1491)

X(1) = cevapoint of X(i) and X(j) for these (i,j):

(2,192), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (50,215), (56,221), (65,73), (244,513)

X(1) = X(i)-cross conjugate of X(j) for these (i,j): (2,87), (3,90), (6,57), (31,19), (33,282), (37,2), (38,75), (42,6), (44,88), (48,63), (55,9), (56,84), (58,267), (65,4), (73,3), (192,43), (207,1490), (221,40), (244,513), (259,258), (266,505), (354,7), (367,366), (500,35), (513,100), (517,80), (518,291), (1491,1492)

X(1) = crosspoint of X(i) and X(j) for these (i,j): (2,7), (8,280), (21,29), (59,110), (75,92), (81,86)

X(1) = crosssum of X(i) and X(j) for these (i,j): (2,192), (4,1148), (6,55), (11,523), (15,202), (16,203), (19,204), (31,48), (34,207), (37,42), (44,678), (50,215), (56,221), (57,1419), (65,73), (214,758), (244,513), (500,942), (512,1015), (774,820), (999,1480)

X(1) = crossdifference of every pair of points on line X(44)X(513)

X(1) = X(i)-Hirst inverse of X(j) for these (i,j): (2,239), (4,243), (6,238), (9,518), (19,240), (57,241), (105,294), (291,292)

X(1) = X(6)-line conjugate of X(44)

X(1) = X(i)-aleph conjugate of X(j) for these (i,j):

(1,1), (2,63), (4,920), (21,411), (29,412), (88,88), (100,100), (162,162), (174,57), (188,40), (190,190), (266,978), (365,43), (366,9), (507,173), (508,169), (513,1052), (651, 651), (653,653), (655,655), (658,658), (660,660), (662,662), (673,673), (771,771), (799,799), (823,823), (897,897)

X(1) = X(i)-beth conjugate of X(j) for these (i,j): (1,56), (2,948), (8,8), (9,45), (21,1), (29,34), (55,869), (99,85), (100,1), (110,603), (162,208), (643,1), (644,1), (663,875), (664,1), (1043,78)

X(1) = insimilicenter of 1st & 2nd Johnson-Yff circles (the exsimilicenter is X(4))

X(1) = orthic-isogonal conjugate of X(46)

X(1) = excentral-isogonal conjugate of X(40)

X(1) = excentral-isotomic conjugate of X(2951)

X(1) = center of Conway circle

X(1) = center of Adams circle

X(1) = X(3) of polar triangle of Conway circle

X(1) = homothetic center of intangents triangle and reflection of extangents triangle in X(3)

X(1) = Hofstadter 1/2 point

X(1) = orthocenter of X(4)X(9)X(885)

X(1) = intersection of tangents at X(7) and X(8) to Lucas cubic K007

X(1) = trilinear product of vertices of 2nd mixtilinear triangle

X(1) = trilinear product of vertices of 2nd Sharygin triangle

X(1) = homothetic center of Mandart-incircle triangle and 2nd isogonal triangle of X(1); see X(36)

X(1) = trilinear pole of the antiorthic axis (which is also the Monge line of the mixtilinear excircles)

X(1) = pole wrt polar circle of trilinear polar of X(92) (line X(240)X(522))

X(1) = X(48)-isoconjugate (polar conjugate) of X(92)

X(1) = X(6)-isoconjugate of X(2)

X(1) = trilinear product of PU(i) for these i: 1, 17, 114, 115, 118, 119, 113

X(1) = barycentric product of PU(i) for these i: 6, 124

X(1) = vertex conjugate of PU(9)

X(1) = bicentric sum of PU(i) for these i: 28, 47, 51, 55, 64

X(1) = trilinear pole of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

X(1) = crossdifference of PU(i) for these i: 33, 50, 57, 58, 74, 76, 78

X(1) = midpoint of PU(i) for these i: 47, 51, 55

X(1) = PU(28)-harmonic conjugate of X(1023)

X(1) = PU(64)-harmonic conjugate of X(351)

X(1) = intersection of diagonals of trapezoid PU(6)PU(31)

X(1) = perspector circumconic centered at X(9)

X(1) = eigencenter of mixtilinear excentral triangle

X(1) = eigencenter of 2nd Sharygin triangle

X(1) = perspector of ABC and unary cofactor triangle of extangents triangle

X(1) = perspector of ABC and unary cofactor triangle of Feuerbach triangle

X(1) = perspector of ABC and unary cofactor triangle of Apollonius triangle

X(1) = perspector of ABC and unary cofactor triangle of 2nd mixtilinear triangle

X(1) = perspector of ABC and unary cofactor triangle of 4th mixtilinear triangle

X(1) = perspector of ABC and unary cofactor triangle of Apus triangle

X(1) = perspector of unary cofactor triangles of 6th and 7th mixtilinear triangles

X(1) = perspector of unary cofactor triangles of 2nd and 3rd extouch triangles

X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 2nd mixtilinear triangle

X(1) = perspector of 5th mixtilinear triangle and unary cofactor triangle of 4th mixtilinear triangle

X(1) = X(3)-of-reflection-triangle-of-X(1)

X(1) = X(1181)-of-2nd-extouch triangle

X(1) = perspector of ABC and orthic-triangle-of-2nd-circumperp-triangle

X(1) = X(4)-of-excentral triangle

X(1) = X(40)-of-Yff central triangle

X(1) = X(20)-of-1st circumperp triangle

X(1) = X(4)-of-2nd circumperp triangle

X(1) = X(4)-of-Fuhrmann triangle

X(1) = X(100)-of-X(1)-Brocard triangle

X(1) = antigonal image of X(80)

X(1) = trilinear pole wrt excentral triangle of antiorthic axis

X(1) = trilinear pole wrt incentral triangle of antiorthic axis

X(1) = Miquel associate of X(7)

X(1) = homothetic center of Johnson triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of 1st Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of 2nd Johnson-Yff triangle and cross-triangle of ABC and 2nd isogonal triangle of X(1)

X(1) = homothetic center of Mandart-incircle triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(1) = homothetic center of medial triangle and cross-triangle of Aquila and anti-Aquila triangles

X(1) = homothetic center of outer Garcia triangle and cross-triangle of Aquila and anti-Aquila triangles

X(1) = X(8)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

X(1) = X(3)-of-Mandart-incircle-triangle

X(1) = X(100)-of-inner-Garcia-triangle

X(1) = Thomson-isogonal conjugate of X(165)

X(1) = X(8)-of-outer-Garcia-triangle

X(1) = X(486)-of-BCI-triangle

X(1) = X(164)-of-orthic-triangle if ABC is acute

X(1) = X(1593)-of-Ascella-triangle

X(1) = excentral-to-Ascella similarity image of X(1697)

X(1) = Dao image of X(1)

X(1) = X(40)-of-reflection of ABC in X(3)

X(1) = radical center of the tangent circles of ABC

X(1) = homothetic center of intangents triangle and anti-tangential midarc triangle

X(1) = K(X(15)) = K(X(16)), as defined at X(174)

X(1) = X(3)-of-hexyl-triangle

X(1) = eigencenter of trilinear obverse triangle of X(2)

X(1) = hexyl-isogonal conjugate of X(40)

X(1) = inverse-in-polar-circle of X(1785)

X(1) = inverse-in-orthoptic-circle-of-Steiner-inellipse of X(5121)

X(1) = inverse-in-OI-inverter of X(1155)

X(1) = inverse-in-Steiner-circumellipse of X(239)

X(1) = inverse-in-MacBeath-circumconic of X(2323)

X(1) = inverse-in-circumconic-centered-at-X(9) of X(44)

X(1) = excentral-to-ABC barycentric image of X(40)

X(1) = excentral-to-ABC functional image of X(164)

X(1) = excentral-to-ABC trilinear image of X(164)

X(1) = orthic-to-ABC functional image of X(4), if ABC is acute

X(1) = orthic-to-ABC trilinear image of X(4), if ABC is acute

X(1) = intouch-to-ABC barycentric image of X(1)

X(1) = excentral-to-intouch similarity image of X(40)

X(1) = ABC-to-excentral barycentric image of X(8)

X(1) = X(1)-vertex conjugate of X(56)

X(1) = perspector of ABC and reflection triangle of intangents triangle

X(1) = perspector of pedal and anticevian triangles of X(40)

X(1) = perspector of hexyl triangle and antipedal triangle of X(40)

X(1) = perspector of hexyl triangle and anticevian triangle of X(57)

X(1) = X(4)-of-Pelletier-triangle



X(2) = CENTROID

Trilinears 1/a : 1/b : 1/cTrilinears bc : ca : abTrilinears csc A : csc B : csc CTrilinears cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos BTrilinears sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec BTrilinears cos A + cos(B - C) : cos B + cos(C - A) : cos C + cos(A - B)Trilinears cos B cos C - cos(B - C) : cos C cos A - cos(C - A) : cos A cos B - cos(A - B)Trilinears tan(A/2) + cot(A/2) : :Trilinears 1 + csc A/2 sin B/2 sin C/2 : :Barycentrics 1 : 1 : 1Tripolars Sqrt[2(b^2 + c^2) - a^2] : :

As a point on the Euler line, X(2) has Shinagawa coefficients (1, 0).

X(2) is the point of concurrence of the medians of ABC, situated 1/3 of the distance from each vertex to the midpoint of the opposite side. More generally, if L is any line in the plane of ABC, then the distance from X(2) to L is the average of the distances from A, B, C to L. An idealized triangular sheet balances atop a pin head located at X(2) and also balances atop any knife-edge that passes through X(2). The triangles BXC, CXA, AXB have equal areas if and only if X = X(2).

If you have The Geometer's Sketchpad, you can view Centroid.

If you have GeoGebra, you can view Centroid.

X(2) is the centroid of the set of 3 vertices, the centroid of the triangle including its interior, but not the centroid of the triangle without its interior; that centroid is X(10).

X(2) is the identity of the group of triangle centers under "barycentric multiplication" defined by

(x : y : z)*(u : v : w) = xu : yv : zw.

X(2) is the unique point X (as a function of a,b,c) for which the vector-sum XA + XB + XC is the zero vector.

Theis here introduced as the circle centered at X(2502) that passes through the isodynamic points, X(15) and X(16). This circle is orthogonal to both the circumcircle and Parry circle. (Randy Hutson, February 10, 2016)

Let A' be the trilinear pole of the perpendicular bisector of BC, and define B' and C' cyclically. A'B'C' is also the anticomplement of the anticomplement of the midheight triangle. A'B'C' is homothetic to the midheight triangle at X(2). (Randy Hutson, January 29, 2018)

Let A'B'C' be the excentral triangle. Let Oa be the A'-McCay circle of triangle A'BC, and define Ob, Oc cyclically. X(2) is the radical center of Oa, Ob, Oc. (Randy Hutson, June 27, 2018)

X(2) is the unique point that is the symmedian point of its antipedal triangle. (Randy Hutson, August 19, 2019)

Let A'B'C' be the midheight triangle. Let A" be the cevapoint of B' and C', and define B" and C" cyclically. The lines AA", BB", CC" concur in X(2). (Randy Hutson, October 8, 2019)

X(2) lies on the Parry circle, Lucas cubic, Thomson cubic, and these lines: 1,8 3,4 6,69 7,9 11,55 12,56 13,16 14,15 17,62 18,61 19,534 31,171 32,83 33,1040 34,1038 35,1479 36,535 37,75 38,244 39,76 40,946 44,89 45,88 51,262 52,1216 54,68 58,540 65,959 66,206 71,1246 72,942 74,113 77,189 80,214 85,241 92,273 94,300 95,97 98,110 99,111 101,116 102,117 103,118 104,119 106,121 107,122 108,123 109,124 112,127 128,1141 129,1298 130,1303 131,1300 133,1294 136,925 137,930 154,1503 165,516 169,1763 174,236 176,1659 178,188 187,316 196,653 201,1393 210,354 216,232 220,1170 222,651 231,1273 242,1851 243,1857 252,1166 253,1073 254,847 257,1432 261,593 265,1511 271,1034 272,284 280,318 283,580 290,327 292,334 294,949 308,702 311,570 314,941 319,1100 322,1108 330,1107 341,1219 351,804 355,944 360,1115 366,367 371,486 372,485 392,517 476,842 480,1223 489,1132 490,1131 495,956 496,1058 514,1022 523,1649 525,1640 561,716 568,1154 572,1746 573,1730 578,1092 585,1336 586,1123 588,1504 589,1505 594,1255 647,850 648,1494 650,693 664,1121 668,1015 670,1084 689,733 743,789 799,873 812,1635 846,1054 914,1442 918,1638 927,1566 954,1260 1073,1249 968,1738 1000,1145 1043,1834 1060,1870 1074,1785 1076,1838 1089,1224 1093,1217 1124,1378 1143,1489 1155,1836 1171,1509 1186,1207 1257,1265 1284,1403 1335,1377 1340,1349 1341,1348 1500,1574 1501,1691 1672,1681 1673,1680 1674,1679 1675,1678 1697,1706 3343,3344 3349,3350 3351,3352

X(2) is the {X(3),X(5)}-harmonic conjugate of X(4). For a list of other harmonic conjugates of X(2), click Tables at the top of this page.

X(2) = midpoint of X(i) and X(j) for these (i,j): {1,3679}, {3,381}, {4,376}, {5,549}, {6,599}, {7,6172}, {8,3241}, {9,6173}, {10,551}, {11,6174}, {13,5463}, {14,5464}, {20,3543}, {21,6175}, {32,7818}, {37,4688}, {39,9466}, {51,3917}, {69,1992}, {75,4664}, {76,7757}, {98,6054}, {99,671}, {110,9140}, {114,6055}, {115,2482}, {125,5642}, {126,9172}, {140,547}, {141,597}, {148,8591}, {154,1853}, {165,1699}, {190,903}, {192,4740}, {210,354}, {329,2094}, {351,9148}, {355,3655}, {373,5650}, {384,7924}, {385,7840}, {392,3753}, {428,7667}, {591,1991}, {618,5459}, {619,5460}, {620,5461}, {631,5071}, {648,1494}, {664,1121}, {668,3227}, {670,3228}, {858,7426}, {1003,7841}, {1086,4370}, {1125,3828}, {1635,4728}, {1638,1639}, {1641,1648}, {1644,1647}, {1649,8371}, {1650,1651}, {2454,2455}, {2479,2480}, {2487,4677}, {2966,5641}, {2976,6161}, {2979,3060}, {3034,3875}, {3034,7292}, {3251,4162}, {3268,9979}, {3448,9143}, {3524,3545}, {3534,3830}, {3576,5587}, {3616,4521}, {3617,3676}, {3623,4468}, {3628,10124}, {3654,3656}, {3681,3873}, {3739,4755}, {3740,3742}, {3817,10164}, {3819,5943}, {3845,8703}, {3929,4654}, {4025,4808}, {4108,5996}, {4120,4750}, {4364,10022}, {4373,4776}, {4379,4893}, {4430,4661}, {4643,4795}, {4730,6332}, {4763,4928}, {5054,5055}, {5108,9169}, {5309,7801}, {5466,9168}, {5485,9741}, {5569,8176}, {5603,5657}, {5640,7998}, {5692,5902}, {5858,5859}, {5860,5861}, {5862,5863}, {5883,10176}, {5891,9730}, {5892,10170}, {5927,10167}, {6032,9829}, {6039,6040}, {6189,6190}, {6545,6546}, {6656,6661}, {6784,6786}, {7615,7618}, {7617,7622}, {7734,10128}, {7753,7810}, {7811,7812}, {7817,7880}, {8010,8011}, {8352,8598}, {8356,8370}, {8360,8368}, {8597,9855}, {8667,9766}, {9185,9191}, {9200,9204}, {9201,9205}, {9761,9763}, {9774,10033}, {9778,9812}, {10162,10163}, {10165,10175}

X(2) = reflection of X(i) in X(j) for these (i,j): (1,551), (3,549), (4,381), (5,547), (6,597), (7,6173), (8,3679), (10,3828), (13,5459), (14,5460), (20,376), (23,7426), (37,4755), (51,5943), (69,599), (75,4688), (76,9466), (98,6055), (99,2482), (100,6174), (110,5642), (111,9172), (115,5461), (140,10124), (144,6172), (145,3241), (147,6054), (148,671), (154,10192), (165,10164), (182,10168), (190,4370), (192,4664), (193,1992), (194,7757), (210,3740), (315,7818), (352,9127), (353,10166), (354,3742), (356,5455), (376,3), (381,5), (384,6661), (547,3628), (549,140), (551,1125), (568,5946), (597,3589), (599,141), (616,5463), (617,5464), (648,3163), (671,115), (903,1086), (944,3655), (1003,8369), (1121,1146), (1278,4740), (1635,4763), (1651,402), (1699,3817), (1962,10180), (1992,6), (2094,57), (2475,6175), (2479,2454), (2480,2455), (2482,620), (2979,3917), (3034,2321), (3060,51), (3091,5071), (3146,3543), (3227,1015), (3228,1084), (3241,1), (3448,9140), (3524,5054), (3534,8703), (3543,4), (3545,5055), (3576,10165), (3617,4521), (3623,3676), (3655,1385), (3676,3616), (3679,10), (3681,210), (3742,3848), (3817,10171), (3828,3634), (3830,3845), (3839,3545), (3845,5066), (3873,354), (3877,392), (3917,3819), (3929,5325), (4240,1651), (4363,10022), (4370,4422), (4430,3873), (4440,903), (4453,1638), (4468,3617), (4521,1698), (4644,4795), (4661,3681), (4664,37), (4669,4745), (4677,4669), (4688,3739), (4728,4928), (4740,75), (4755,4698), (4776,3161), (4795,4670), (4808,3239), (4808,8834), (5066,10109), (5071,1656), (5309,7817), (5459,6669), (5460,6670), (5461,6722), (5463,618), (5464,619), (5466,8371), (5468,1641), (5569,1153), (5587,10175), (5603,5886), (5640,373), (5642,5972), (5692,10176), (5731,3576), (5860,591), (5861,1991), (5862,5858), (5863,5859), (5883,3833), (5890,9730), (5891,10170), (5902,5883), (5918,10178), (5919,10179), (5927,10157), (5943,6688), (6031,9829), (6032,10162), (6054,114), (6055,6036), (6161,2505), (6172,9), (6173,142), (6174,3035), (6175,442), (6546,10196), (6655,7924), (6661,7819), (6688,10219), (6792,9169), (7426,468), (7615,7617), (7618,7622), (7620,7615), (7622,7619), (7671,10177), (7757,39), (7779,7840), (7801,7880), (7811,7810), (7812,7753), (7818,626), (7833,8356), (7840,325), (7924,6656), (7998,5650), (8182,5569), (8353,8354), (8354,8358), (8356,8359), (8368,8365), (8369,8368), (8591,99), (8596,148), (8597,8352), (8860,3054), (9123,9125), (9140,125), (9143,110), (9144,5465), (9147,351), (9168,1649), (9172,6719), (9185,9189), (9263,3227), (9466,3934), (9485,9123), (9730,5892), (9778,165), (9779,7988), (9812,1699), (9829,10163), (9855,8598), (9909,10154), (9939,7811), (9965,2094), (9979,1637), (10022,4472), (10056,10197), (10072,10199), (10162,10173), (10166,10160), (10175,10172)

X(2) = isogonal conjugate of X(6)

X(2) = isotomic conjugate of X(2)

X(2) = cyclocevian conjugate of X(4)

X(2) = circumcircle-inverse of X(23)

X(2) = Conway-circle-inverse of X(38473)

X(2) = nine-point-circle-inverse of X(858)

X(2) = Brocard-circle-inverse of X(110)

X(2) = complement of X(2)

X(2) = anticomplement of X(2)

X(2) = anticomplementary conjugate of X(69)

X(2) = complementary conjugate of X(141)

X(2) = insimilicenter of incircle and Spieker circle

X(2) = insimilicenter of incircle and AC-incircle

X(2) = exsimilicenter of Spieker circle and AC-incircle

X(2) = insimilicenter of Conway circle and Spieker radical circle

X(2) = insimilicenter of polar circle and de Longchamps circle

X(2) = harmonic center of pedal circles of X(13) and X(14) (which are also the pedal circles of X(15) and X(16))

X(2) = X(99)-of -1st-Parry-triangle

X(2) = X(98)-of-2nd-Parry-triangle

X(2) = X(2)-of-1st-Brocard-triangle

X(2) = X(111)-of-4th-Brocard-triangle

X(2) = X(110)-of-X(2)-Brocard-triangle

X(2) = X(110)-of-orthocentroidal-triangle

X(2) = X(353)-of-circumsymmedial-triangle

X(2) = X(165)-of-orthic-triangle if ABC is acute

X(2) = X(51)-of-excentral-triangle

X(2) = inverse-in-polar-circle of X(468)

X(2) = inverse-in-de-Longchamps-circle of X(858)

X(2) = inverse-in-MacBeath-circumconic of X(323)

X(2) = inverse-in-Feuerbach-hyperbola of X(390)

X(2) = inverse-in-circumconic-centered-at-X(1) of X(3935)

X(2) = inverse-in-circumconic-centered-at-X(9) of X(3218)

X(2) = inverse-in-excircles-radical-circle of X(5212)

X(2) = inverse-in-Parry-isodynamic-circle of X(353)

X(2) = barycentric product of (real or nonreal) circumcircle intercepts of the de Longchamps line

X(2) = barycentric product of circumcircle intercepts of line X(325)X(523)

X(2) = barycentric product of PU(3)

X(2) = barycentric product of PU(35)

X(2) = harmonic center of nine-point circle and Johnson circle

X(2) = pole wrt polar circle of trilinear polar of X(4) (orthic axis)

X(2) = polar conjugate of X(4)

X(2) = excentral-to-ABC functional image of X(165)

X(2) = excentral-to-ABC barycentric image of X(165)

X(2) = orthic-to-ABC functional image of X(51)

X(2) = orthic-to-ABC barycentric image of X(51)

X(2) = incentral-to-ABC functional image of X(1962)

X(2) = incentral-to-ABC barycentric image of X(1962)

X(2) = Feuerbach-to-ABC functional image of X(5947)

X(2) = Feuerbach-to-ABC barycentric image of X(5947)

X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle

X(2) = trilinear pole, wrt orthocentroidal triangle, of Fermat axis

X(2) = trilinear pole, wrt 1st Parry triangle, of line X(1499)X(8598)

X(2) = pole of Brocard axis wrt Stammler hyperbola

X(2) = pole of de Longchamps line wrt the nine-point circle

X(2) = pole of de Longchamps line wrt the de Longchamps circle

X(2) = pole of orthic axis wrt polar circle

X(2) = crosspoint of X(3) and X(6) wrt both the excentral and tangential triangles

X(2) = intersection of tangents at X(1) and X(9) to the hyperbola passing through X(1), X(9) and the excenters (the Jerabek hyperbola of the excentral triangle)

X(2) = crosspoint of X(1) and X(9) wrt excentral triangle

X(2) = crosspoint of X(3) and X(6) wrt excentral triangle

X(2) = crosspoint of X(7) and X(8) wrt 2nd Conway triangle

X(2) = antipode of X(3228) in hyperbola {{A,B,C,X(2),X(6)}}

X(2) = antipode of X(1494) in hyperbola {{A,B,C,X(2),X(69)}}

X(2) = perspector of pedal and anticevian triangles of X(20)

X(2) = homothetic center of the 2nd pedal triangle of X(4) and the 3rd pedal triangle of X(3)

X(2) = perspector of ABC and the reflection in X(6) of the pedal triangle of X(6)

X(2) = perspector of orthic triangle and polar triangle of the complement of the polar circle

X(2) = Moses-radical-circle-inverse of X(34235)

X(2) = X(6374)-cross conjugate of X(194)

X(2) = 1st-Brocard-isogonal conjugate of X(3734)

X(2) = X(i)-Ceva conjugate of X(j) for these (i,j): (1,192), (4,193), (6,194), (7,145), (8,144), (30,1494), (69,20), (75,8), (76,69), (83,6), (85,7), (86,1), (87,330), (95,3), (98,385), (99,523), (190,514), (264,4), (274,75), (276, 264), (287,401), (290,511), (308,76), (312,329), (325,147), (333,63), (348,347), (491,487), (492,488), (523,148), (626,1502)

X(2) = cevapoint of X(i) and X(j) for these (i,j): (1,9), (3,6), (5,216), (10,37), (11,650), (32,206), (39,141), (44,214), (57,223), (114,230), (115,523), (125,647), (128,231), (132,232), (140,233), (188,236), (5408,5409)

X(2) = X(i)-cross conjugate of X(j) for these (i,j):

(1,7), (3,69), (4,253), (5,264), (6,4), (9,8), (10,75), (32,66), (37,1), (39,6), (44,80), (57,189), (75,330), (114,325), (140,95), (141,76), (142,85), (178,508), (187,67), (206,315), (214,320), (216,3), (223,329), (226,92), (230,98), (233,5), (281,280), (395,14), (396,13), (440,306), (511,290), (514,190), (523,99)

X(2) = crosspoint of X(i) and X(j) for these (i,j): (1,87), (75,85), (76,264), (83,308), (86,274), (95,276),(36308,36311)

X(2) = crosssum of X(i) and X(j) for these (i,j): (1,43), (2,194), (31,41), (32,184), (42,213), (51,217), (125,826), (649,1015), (688,1084), (902,1017), (1400,1409)

X(2) = crossdifference of every pair of points on line X(187)X(237)

X(2) = X(i)-Hirst inverse of X(j) for these (i,j):

(1,239), (3,401), (4,297), (6,385), (21,448), (27,447), (69,325), (75,350), (98,287), (115,148), (193,230), (291,335), (298,299), (449,452)

X(2) = X(3)-line conjugate of X(237) = X(316)-line conjugate of X(187)

X(2) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1045), (2,191), (86,2), (174,1046), (333,20), (366,846)

X(2) = X(i)-beth conjugate of X(j) for these (i,j): (2,57), (21,995) (190,2), (312,312), (333,2), (643,55), (645,2), (646,2), (648,196), (662,222)

X(2) = one of two harmonic traces of the power circles; the other is X(858)

X(2) = one of two harmonic traces of the McCay circles; the other is X(111)

X(2) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (4,6,1640), (4,10,4040)

X(2) = centroid of PU(1)X(76) (1st, 2nd, 3rd Brocard points)

X(2) = trilinear pole of PU(i) for these i: 24, 41

X(2) = crossdifference of PU(i) for these i: 2, 26

X(2) = trilinear product of PU(i) for these i: 6,124

X(2) = bicentric sum of PU(i) for these i: 116, 117, 118, 119, 138, 148

X(2) = midpoint of PU(i) for these i: 116, 117, 118, 119, 127

X(2) = intersection of diagonals of trapezoid PU(11)PU(45) (lines P(11)P(45) and U(11)U(45))

X(2) = X(5182) of 6th Brocard triangle (see X(384))

X(2) = PU(148)-harmonic conjugate of X(669)

X(2) = bicentric difference of PU(147)

X(2) = eigencenter of 2nd Brocard triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas central triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) central triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas tangents triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) tangents triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas inner triangle

X(2) = perspector of ABC and unary cofactor triangle of Lucas(-1) inner triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st anti-Brocard triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st Sharygin triangle

X(2) = perspector of ABC and unary cofactor triangle of 2nd Sharygin triangle

X(2) = perspector of ABC and unary cofactor triangle of 1st Pamfilos-Zhou triangle

X(2) = perspector of ABC and unary cofactor triangle of Artzt triangle

X(2) = perspector of 1st Parry triangle and unary cofactor of 3rd Parry triangle

X(2) = X(6032) of 4th anti-Brocard triangle

X(2) = orthocenter of X(3)X(9147)X(9149)

X(2) = exsimilicenter of Artzt and anti-Artzt circles; the insimilicenter is X(183)

X(2) = perspector of ABC and cross-triangle of inner- and outer-squares triangles

X(2) = perspector of ABC and 2nd Brocard triangle of 1st Brocard triangle

X(2) = perspector of half-altitude triangle and cross-triangle of ABC and half-altitude triangle

X(2) = 4th-anti-Brocard-to-anti-Artzt similarity image of X(111)

X(2) = homothetic center of Aquila triangle and cross-triangle of Aquila and anti-Aquila triangles

X(2) = X(551)-of-cross-triangle-of Aquila-and-anti-Aquila-triangles

X(2) = harmonic center of polar circle and circle O(PU(4))

X(2) = Thomson-isogonal conjugate of X(3)

X(2) = Lucas-isogonal conjugate of X(20)

X(2) = X(3679)-of-outer-Garcia-triangle

X(2) = Dao image of X(13)

X(2) = Dao image of X(14)

X(2) = center of equilateral triangle X(3)PU(5)

X(2) = center of equilateral triangle formed by the circumcenters of BCF, CAF, ABF, where F = X(13)

X(2) = center of equilateral triangle formed by the circumcenters of BCF', CAF', ABF', where F' = X(14)

X(2) = trisector nearest X(5) of segment X(3)X(5)

X(2) = trisector nearest X(4) of segment X(4)X(20)

X(2) = pedal antipodal perspector of X(15)

X(2) = pedal antipodal perspector of X(16)

X(2) = K(X(3)), as defined at X(174)

X(2) = Ehrmann-mid-to-Johnson similarity image of X(381)

X(2) = Kiepert hyperbola antipode of X(671)

X(2) = antigonal conjugate of X(671)

X(2) = trilinear square of X(366)

X(2) = intersection of diagonals of trapezoid X(1)X(7)X(8)X(9)

X(2) = Danneels point of X(99)

X(2) = Danneels point of X(648)

X(2) = perspector of Spieker circle

X(2) = orthic-isogonal conjugate of X(193)

X(2) = X(154)-of-intouch-triangle

X(2) = Vu circlecevian point V(X(13),X(14))



X(3) = CIRCUMCENTER

Trilinears cos A : cos B : cos CTrilinears a(b+ c- a) : b(c+ a- b) : c(a+ b- cBarycentrics sin 2A : sin 2B : sin 2CBarycentrics tan B + tan C : tan C + tan A: tan A + tan BBarycentrics S^2 - SB SC : :Barycentrics 1 - cot B cot C : :Tripolars 1 : 1 : 1

As a point on the Euler line, X(3) has Shinagawa coefficients (1, -1).

X(3) is the point of concurrence of the perpendicular bisectors of the sides of ABC. The lengths of segments AX, BX, CX are equal if and only if X = X(3). This common distance is the radius of the circumcircle, which passes through vertices A,B,C. Called the circumradius, it is given by R = a/(2 sin A) = abc/(4*area(ABC)).

The tangents at vertices of excentral triangle to the McCay cubic K003 concur in X(3). Also, the tangents at A,B,C to the orthocubic K006 concur in X(3). (Randy Hutson, November 18, 2015)

Let A'B'C' be the cevian triangle of X(4). Let A" be X(4)-of-AB'C', and define B", C" cyclically. The lines AA", BB", CC" concur in X(3). (Randy Hutson, June 27, 2018)

Let P be a point in the plane of ABC. Let P' be the isogonal conjugate of P. Let P" be the pedal antipodal perspector of P. X(3) is the unique point P for which P' = P". (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X(3) is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X(3) gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X(4) has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X(4) is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X(4) gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

If you have The Geometer's Sketchpad, you can view Circumcenter.

If you have GeoGebra, you can view Circumcenter.

Let T be any one of these trianges: {Aries, X(3)-Ehrmann, X3-ABC reflections, 3rd pedal of X(3), 3rd antipedal of X(3), inner-Le Viet An, outer-Le Viet An}. Let O A be the circle centered at the A-vertex of T and passing through A; define O B and O C cyclically. X(3) is the radical center of O A , O B , O C . (Randy Hutson, August 30, 2020)

X(3) lies on the Thomson cubic, the Darboux cubic, the Napoleon cubic, the Neuberg cubic, the McCay cubic, then Darboux quintic, and these lines: {1,35}, {2,4}, {6,15}, {7,943}, {8,100}, {9,84}, {10,197}, {11,499}, {12,498}, {13,17}, {14,18}, {19,1871}, {31,601}, {33,1753}, {34,1465}, {37,975}, {38,976}, {41,218}, {42,967}, {43,5247}, {47,1399}, {48,71}, {49,155}, {51,3527}, {54,97}, {60,1175}, {63,72}, {64,154}, {66,141}, {67,542}, {68,343}, {69,332}, {73,212}, {74,110}, {76,98}, {77,1410}, {80,5445}, {81,5453}, {83,262}, {85,5088}, {86,1246}, {90,1898}, {95,264}, {96,5392}, {101,103}, {102,109}, {105,277}, {106,1293}, {107,1294}, {108,1295}, {111,1296}, {112,1297}, {113,122}, {114,127}, {115,2079}, {119,123}, {125,131}, {128,1601}, {142,516}, {143,1173}, {144,5843}, {145,1483}, {147,2896}, {149,1484}, {158,243}, {161,1209}, {164,3659}, {169,910}, {172,2276}, {191,1768}, {193,1353}, {194,385}, {200,963}, {201,1807}, {207,1767}, {214,2800}, {217,3289}, {223,1035}, {225,1074}, {226,4292}, {227,1455}, {230,2549}, {232,1968}, {238,978}, {248,3269}, {252,930}, {256,987}, {269,939}, {295,2196}, {296,820}, {298,617}, {299,616}, {302,621}, {303,622}, {305,1799}, {315,325}, {323,3431}, {329,2096}, {345,1791}, {347,1119}, {348,1565}, {351,2780}, {352,353}, {356,3278}, {358,6120}, {373,3066}, {380,2257}, {388,495}, {390,1058}, {392,3420}, {393,1217}, {395,398}, {396,397}, {476,477}, {480,5223}, {485,590}, {486,615}, {489,492}, {490,491}, {496,497}, {501,5127}, {513,3657}, {518,3433}, {519,3654}, {523,5664}, {524,5486}, {525,878}, {528,3813}, {532,5859}, {533,5858}, {539,3519}, {541,5642}, {543,5569}, {551,3653}, {595,995}, {604,2269}, {607,1951}, {608,1950}, {609,5280}, {611,1469}, {612,5322}, {613,1428}, {614,5310}, {618,635}, {619,636}, {623,629}, {624,630}, {639,641}, {640,642}, {653,1148}, {659,2826}, {662,1098}, {667,1083}, {669,1499}, {690,6334}, {691,842}, {692,2807}, {695,1613}, {732,6308}, {741,6010}, {758,5884}, {759,6011}, {805,2698}, {840,2742}, {843,2709}, {846,2944}, {847,925}, {895,4558}, {901,953}, {902,1201}, {905,1946}, {915,2969}, {917,1305}, {920,1858}, {927,2724}, {929,2723}, {934,972}, {935,2697}, {938,3488}, {945,1457}, {947,5399}, {950,1210}, {951,1407}, {955,1170}, {960,997}, {962,1621}, {968,6051}, {974,5504}, {984,3497}, {1000,1476}, {1014,3945}, {1015,2241}, {1018,4513}, {1033,1249}, {1037,1066}, {1046,4650}, {1047,2636}, {1054,1283}, {1055,1334}, {1056,3600}, {1057,1450}, {1069,6238}, {1072,3011}, {1075,1941}, {1093,1105}, {1104,3752}, {1107,4386}, {1124,2066}, {1131,3316}, {1132,3317}, {1135,6121}, {1137,6122}, {1138,3471}, {1139,3370}, {1140,3397}, {1167,1413}, {1177,1576}, {1180,1627}, {1184,1194}, {1196,1611}, {1199,1994}, {1203,5313}, {1211,5810}, {1213,5816}, {1247,2640}, {1263,3459}, {1270,5874}, {1271,5875}, {1276,5240}, {1277,5239}, {1290,2687}, {1298,1303}, {1301,5897}, {1304,2693}, {1308,2717}, {1309,2734}, {1330,4417}, {1331,1797}, {1335,2067}, {1337,3489}, {1338,3490}, {1348,2040}, {1349,2039}, {1364,1795}, {1386,3941}, {1389,2320}, {1397,1682}, {1398,1870}, {1400,2268}, {1406,1464}, {1411,1772}, {1412,2213}, {1425,3561}, {1427,1448}, {1433,2188}, {1445,5728}, {1446,3188}, {1447,3673}, {1452,1905}, {1453,2999}, {1471,2293}, {1475,2280}, {1495,3426}, {1500,2242}, {1506,5475}, {1568,3521}, {1575,4426}, {1587,3068}, {1588,3069}, {1602,2550}, {1603,2551}, {1612,4000}, {1625,1987}, {1630,3197}, {1632,2790}, {1633,5698}, {1661,2883}, {1672,3238}, {1673,3237}, {1676,5403}, {1677,5404}, {1696,3731}, {1698,4413}, {1699,3624}, {1709,3683}, {1714,5721}, {1723,2264}, {1724,3216}, {1728,1864}, {1737,1837}, {1745,1935}, {1762,2939}, {1770,1836}, {1779,1780}, {1788,3486}, {1794,3173}, {1796,3690}, {1808,4173}, {1810,4587}, {1811,4571}, {1813,3270}, {1834,5292}, {1901,5747}, {1914,2275}, {1916,3406}, {1918,2274}, {1939,6181}, {1960,2821}, {1986,2904}, {2007,3235}, {2008,3236}, {2053,2108}, {2120,3463}, {2121,3482}, {2130,3343}, {2131,3350}, {2133,5670}, {2163,2334}, {2174,2911}, {2183,2267}, {2197,2286}, {2222,2716}, {2292,3724}, {2329,3501}, {2346,3296}, {2407,2452}, {2548,3815}, {2688,2690}, {2689,2695}, {2691,2752}, {2692,2758}, {2694,2766}, {2696,2770}, {2699,2703}, {2700,2702}, {2701,2708}, {2704,2711}, {2705,2712}, {2706,2713}, {2707,2714}, {2710,2715}, {2718,2743}, {2719,2744}, {2720,2745}, {2721,2746}, {2722,2747}, {2725,2736}, {2726,2737}, {2727,2738}, {2728,2739}, {2729,2740}, {2730,2751}, {2731,2757}, {2732,2762}, {2733,2765}, {2735,2768}, {2783,4436}, {2792,4655}, {2797,6130}, {2801,3678}, {2810,3939}, {2814,3960}, {2827,4491}, {2854,5505}, {2886,4999}, {2888,3448}, {2916,3456}, {2951,3646}, {2971,3563}, {3006,5300}, {3058,4309}, {3061,3496}, {3065,3467}, {3092,5413}, {3093,5412}, {3100,6198}, {3101,6197}, {3165,5669}, {3166,5668}, {3177,3732}, {3200,3205}, {3201,3206}, {3218,3418}, {3219,3876}, {3224,6234}, {3229,3360}, {3272,3334}, {3276,3280}, {3277,3282}, {3305,5927}, {3306,5439}, {3332,4648}, {3341,3347}, {3351,3354}, {3366,3391}, {3367,3392}, {3373,3387}, {3374,3388}, {3381,5402}, {3382,5401}, {3399,3407}, {3413,6178}, {3414,6177}, {3417,3869}, {3436,5552}, {3437,5224}, {3440,5674}, {3441,5675}, {3447,6328}, {3452,6259}, {3460,3465}, {3461,3483}, {3462,5667}, {3464,3466}, {3474,3485}, {3555,3870}, {3582,4330}, {3583,4324}, {3584,4325}, {3585,4316}, {3589,5480}, {3614,5326}, {3620,5921}, {3632,5288}, {3647,3652}, {3667,4057}, {3679,5258}, {3681,4420}, {3687,5814}, {3694,5227}, {3705,5015}, {3710,3977}, {3711,5531}, {3733,6003}, {3734,3934}, {3740,5302}, {3824,5715}, {3849,6232}, {3874,4973}, {3877,4881}, {3889,3957}, {3901,4880}, {3925,6253}, {4001,4101}, {4317,4995}, {4338,4870}, {4340,5323}, {4549,4846}, {4653,6176}, {4720,5372}, {4850,5262}, {4993,4994}, {5226,5714}, {5260,5818}, {5268,5345}, {5275,5277}, {5284,5550}, {5286,5305}, {5306,5319}, {5346,5355}, {5436,5437}, {5441,5442}, {5443,5444}, {5530,5725}, {5541,6264}, {5590,5594}, {5591,5595}, {5606,5951}, {5638,6141}, {5639,6142}, {5640,5643}, {5656,6225}, {5658,5811}, {5672,6191}, {5673,6192}, {5735,6173}, {5962,5963}, {5971,6031}, {6082,6093}, {6118,6250}, {6119,6251}, {6228,6230}, {6229,6231}, {6233,6323}, {6236,6325}, {6294,6295}, {6296,6298}, {6297,6299}, {6300,6302}, {6301,6303}, {6304,6306}, {6305,6307}, {6311,6313}, {6312,6314}, {6315,6317}, {6316,6318}, {6391,6461}, {6413,6458}, {6414,6457}, {6581,6582}

X(3) is the {X(2),X(4)}-harmonic conjugate of X(5). For a list of other harmonic conjugates of X(3), click Tables at the top of this page. If triangle ABC is acute, then X(3) is the incenter of the tangential triangle and the Bevan point, X(40), of the orthic triangle.

X(3) = midpoint of X(i) and X(j) for these (i,j): (1,40), (2,376), (4,20), (22,378), (74,110), (98,99), (100,104), (101,103), (102,109), (476,477)

X(3) = reflection of X(i) in X(j) for these (i,j): (1,1385), (2,549), (4,5), (5,140), (6,182), (20,550), (52,389), (110,1511), (114,620), (145,1483), (149,1484), (155,1147), (193,1353), (195,54), (265,125), (355,10), (381,2), (382,4), (399,110), (550,548), (576,575), (946,1125), (1351,6), (1352,141), (1482,1)

X(3) = isogonal conjugate of X(4)

X(3) = isotomic conjugate of X(264)

X(3) = complement of X(4)

X(3) = anticomplement of X(5)

X(3) = complementary conjugate of X(5)

X(3) = anticomplementary conjugate of X(2888)

X(3) = nine-point-circle-inverse of X(2072)

X(3) = orthocentroidal-circle-inverse of X(5)

X(3) = 1st-Lemoine-circle-inverse of X(2456)

X(3) = 2nd-Lemoine-circle-inverse of X(1570)

X(3) = Conway-circle-inverse of X(38474)

X(3) = eigencenter of the medial triangle

X(3) = eigencenter of the tangential triangle

X(3) = exsimilicenter of 1st and 2nd Kenmotu circles

X(3) = exsimilicenter of nine-point circle and tangential circle

X(3) = X(1)-of-Trinh-triangle if ABC is acute

X(3) = X(i)-Ceva conjugate of X(j) for these (i,j): (2,6), (4,155), (5,195), (20,1498), (21,1), (22,159), (30,399), (63,219), (69,394), (77,222), (95,2), (96,68), (99,525), (100,521), (110,520), (250, 110), (283,255)

X(3) = cevapoint of X(i) and X(j) for these (i,j): (6,154), (48,212), (55,198), (71,228), (185,417), (216,418)

X(3) = X(i)-cross conjugate of X(j) for these (i,j): (48,222), (55,268), (71,63), (73,1), (184,6), (185,4), (212,219), (216,2), (228,48), (520,110)

X(3) = crosspoint of X(i) and X(j) for these (i,j): (1,90), (2,69), (4,254), (21,283), (54,96), (59,100), (63,77), (78,271), (81,272), (95,97), (99,249), (110,250), (485,486)



X(3) = crosssum of X(i) and X(j) for these (i,j):

(1,46), (2,193), (3,155), (5,52), (6,25), (11,513), (19,33), (30,113), (37,209), (39, 211), (51,53), (65,225), (114,511), (115,512), (116,514), (117, 515), (118,516), (119,517), (120,518), (121,519), (122,520), (123,521), (124,522), (125,523), (126,524), (127,525), (128,1154), (136,924), (184,571), (185,235), (371,372), (487,488)

X(3) = crossdifference of every pair of points on the line X(230)X(231)

X(3) = X(i)-Hirst inverse of X(j) for these (i,j): (2, 401), (4,450), (6,511), (21,416), (194, 385)

X(3) = X(2)-line conjugate of X(468)

X(3) = X(i)-aleph conjugate of X(j) for these (i,j): (1,1046), (21,3), (188,191), (259,1045)

X(3) = X(i)-beth conjugate of X(j) for these (i,j): (3,603), (8,355), (21,56), (78,78), (100,3), (110,221), (271,84), (283,3), (333,379), (643,8)

X(3) = center of inverse-in-de-Longchamps-circle-of-anticomplementary-circle

X(3) = perspector of inner and outer Napoleon triangles

X(3) = Hofstadter 2 point

X(3) = trilinear product of vertices of 2nd Brocard triangle

X(3) = orthocenter of X(i)X(j)X(k) for these (i,j,k): (1,8,5556), (1,9,885), (2,6,1640), (2,10,4049), (3,6,879), (3,66,2435), (4,6,879), (7,8,885), (67,74,879), (6,64,2435), (4,66,2435)

X(3) = intersection of tangents at X(3) and X(4) to Orthocubic K006

X(3) = homothetic center of tangential triangle and 2nd isogonal triangle of X(4); see X(36)

X(3) = trilinear pole of line X(520)X(647)

X(3) = crossdifference of PU(4)

X(3) = trilinear product of PU(16)

X(3) = barycentric product of PU(22)

X(3) = midpoint of PU(i) for these i: 37, 44

X(3) = bicentric sum of PU(i) for these i: 37, 44, 63, 125

X(3) = vertex conjugate of PU(39)

X(3) = PU(63)-harmonic conjugate of X(351)

X(3) = PU(125)-harmonic conjugate of X(650)

X(3) = intersection of tangents to orthocentroidal circle at PU(5)

X(3) = X(3398) of 5th Brocard triangle (see X(32))

X(3) = X(182) of 6th Brocard triangle (see X(384))

X(3) = homothetic center of 1st anti-Brocard triangle and 6th Brocard triangle

X(3) = similitude center of antipedal triangles of the 1st and 2nd Brocard points (PU(1))

X(3) = inverse-in-polar-circle of X(403)

X(3) = inverse-in-{circumcircle, nine-point circle}-inverter of X(858)

X(3) = inverse-in-de-Longchamps-circle of X(3153)

X(3) = inverse-in-Steiner-circumellipse of X(401)

X(3) = inverse-in-Steiner-inellipse of X(441)

X(3) = inverse-in-MacBeath-circumconic of X(3284)

X(3) = radical trace of circumcircle and 8th Lozada circle

X(3) = perspector of medial triangle and polar triangle of the complement of the polar circle

X(3) = pole of line X(6)X(110) wrt Parry circle

X(3) = pole wrt polar circle of trilinear polar of X(2052) (line X(403)X(523))

X(3) = pole wrt {circumcircle, nine-point circle}-inverter of de Longchamps line

X(3) = polar conjugate of X(2052)

X(3) = X(i)-isoconjugate of X(j) for these (i,j): (6,92), (24,91), (25,75), (48,2052), (93,2964), (112,1577), (1101,2970), (2962,3518)

X(3) = X(30)-vertex conjugate of X(523)

X(3) = homothetic center of any 2 of {tangential, Kosnita, 2nd Euler} triangles

X(3) = X(5)-of-excentral-triangle

X(3) = X(26)-of-intouch-triangle

X(3) = antigonal image of X(265)

X(3) = X(2)-of-antipedal-triangle-of-X(6)

X(3) = perspector of the MacBeath Circumconic

X(3) = perspector of ABC and unary cofactor triangle of 5th Euler triangle

X(3) = intersection of trilinear polars of any 2 points on the MacBeath circumconic

X(3) = circumcevian isogonal conjugate of X(1)

X(3) = orthology center of ABC and orthic triangle

X(3) = orthology center of Fuhrmann triangle and ABC

X(3) = orthic isogonal conjugate of X(155)

X(3) = Miquel associate of X(2)

X(3) = X(40)-of-orthic-triangle if ABC is acute

X(3) = X(98)-of-1st-Brocard-triangle

X(3) = X(1380)-of-2nd-Brocard-triangle

X(3) = X(399)-of-orthocentroidal-triangle

X(3) = X(104)-of X(1)-Brocard-triangle

X(3) = X(74)-of X(2)-Brocard-triangle

X(3) = X(74)-of-X(4)-Brocard-triangle

X(3) = X(597)-of-antipedal-triangle-of-X(2)

X(3) = X(182)-of-1st-anti-Brocard-triangle

X(3) = X(381)-of-4th-anti-Brocard-triangle

X(3) = QA-P12 (Orthocenter of the QA-Diagonal Triangle)-of-quadrilateral X(98)X(99)X(110)X(111)

X(3) = orthocenter of X(2)X(9147)X(9149)

X(3) = perspector of ABC and 1st Brocard triangle of 6th Brocard triangle

X(3) = perspector of ABC and 1st Brocard triangle of circumorthic triangle

X(3) = perspector of ABC and 1st Brocard triangle of dual of orthic triangle

X(3) = perspector of ABC and cross-triangle of ABC and half-altitude triangle

X(3) = homothetic center of inner Yff triangle and cross-triangle of ABC and 1st Johnson-Yff triangle

X(3) = homothetic center of outer Yff triangle and cross-triangle of ABC and 2nd Johnson-Yff triangle

X(3) = anti-Artzt-to-4th-anti-Brocard similarity image of X(6)

X(3) = Thomson-isogonal conjugate of X(2)

X(3) = Lucas-isogonal conjugate of X(2979)

X(3) = X(4)-of-2nd-anti-extouch triangle

X(3) = X(185)-of-A'B'C', as described in ADGEOM #2697 (8/26/2015, Tran Quang Hung)

X(3) = X(5)-of-3rd-anti-Euler-triangle

X(3) = X(5)-of-4th-anti-Euler-triangle

X(3) = X(671)-of-McCay-triangle

X(3) = Dao image of X(4)

X(3) = centroid of ABCX(20)

X(3) = Kosnita(X(20),X(2)) point

X(3) = centroid of incenter and excenters

X(3) = X(265)-of-Fuhrmann-triangle

X(3) = intersection of tangents to 2nd Lemoine circle at intersections with Brocard circle

X(3) = perspector of ABC and antipedal triangle of X(64)

X(3) = trisector nearest X(5) of segment X(5)X(20)

X(3) = Ehrmann-vertex-to-Ehrmann-side similarity image of X(4)

X(3) = Ehrmann-mid-to-ABC similarity image of X(4)

X(3) = Ehrmann-mid-to-Johnson similarity image of X(5)

X(3) = Johnson-to-Ehrmann-mid similarity image of X(20)

X(3) = center of inverse similitude of AAOA triangle and Ehrmann side-triangle

X(3) = X(5)-of-hexyl-triangle

X(3) = X(175)-of-Lucas-central-triangle

X(3) = reflection of X(2080) in the Lemoine axis

X(3) = excentral-isogonal conjugate of X(191)

X(3) = excentral-isotomic conjugate of X(2938)

X(3) = crosssum of foci of orthic inconic

X(3) = crosspoint of foci of orthic inconic

X(3) = similicenter of antipedal triangles of PU(1)

X(3) = excentral-to-ABC functional image of X(40)

X(3) = orthic-to-ABC barycentric image of X(4)

X(3) = orthic-to-ABC functional image of X(5)

X(3) = Feuerbach-to-ABC functional image of X(5)

X(3) = intouch-to-ABC functional image of X(1)

X(3) = ABC-to-excentral barycentric image of X(10)

X(3) = concurrence of Euler lines of intouch triangle and A-, B-, and C-extouch triangles

X(3) = perspector of hexyl triangle and cevian triangle of X(21)

X(3) = perspector of pedal and anticevian triangles of X(1498)

X(3) = perspector of ABC and medial triangle of pedal triangle of X(20)

X(3) = perspector of ABC and the reflection in X(6) of the antipedal triangle of X(6)

X(3) = tangential-isotomic conjugate of tangential-isogonal conjugate of X(35225)

X(3) = Moses-radical-circle-inverse of X(35901)

X(3) = 1st-Brocard-isogonal conjugate of X(2782)

X(3) = {X(i),X(j)}-harmonic conjugate of X(k) for these (i,j,k): (1,35,55), (1,36,56), (1,46,65), (1,55,3295), (1,56,999), (1,57,942), (1,165,40), (1,171,5711), (1,484,5903), (1,1038,1060), (1,1040,1062), (1,1754,5706), (1,2093,3340), (1,3333,5045), (1,3336,5902), (1,3338,354), (1,3361,3333), (1,3550,5255), (1,3576,1385), (1,3612,2646), (1,3746,3303), (1,5010,35), (1,5119,3057), (1,5131,3336), (1,5264,5710), (1,5563,3304), (1,5697,2098), (1,5903,2099), (2,4,5), (2,5,1656), (2,20,4), (2,21,405), (2,22,25), (2,23,1995), (2,24,6642), (2,25,5020), (2,140,3526), (2,186,6644), (2,377,442), (2,381,5055), (2,382,3851), (2,401,458), (2,404,474), (2,411,3149), (2,418,6638), (2,452,5084), (2,464,440), (2,546,5079), (2,548,1657), (2,549,5054), (2,550,382), (2,631,140), (2,858,5094), (2,859,4245), (2,1010,2049), (2,1113,1344), (2,1114,1345), (2,1370,427), (2,1599,1583), (2,1600,1584), (2,1656,5070), (2,1657,3843), (2,2071,378), (2,2475,2476), (2,2478,4187), (2,2554,2570), (2,2555,2571), (2,2675,2676), (2,3090,3628), (2,3091,3090), (2,3146,3091), (2,3151,469), (2,3152,5125), (2,3522,20), (2,3523,631), (2,3524,549), (2,3525,632), (2,3528,550), (2,3529,546), (2,3534,3830), (2,3543,3545), (2,3545,547), (2,3546,3548), (2,3547,3549), (2,3548,6640), (2,3549,6639), (2,3552,384), (2,3627,5072), (2,3832,5056), (2,3839,5071), (2,4184,1011), (2,4188,404), (2,4189,21), (2,4190,377), (2,4210,4191), (2,4216,859), (2,4226,1316), (2,5046,4193), (2,5056,5067), (2,5059,3832), (2,5189,5169), (2,6636,22), (4,5,381), (4,21,3560), (4,24,25), (4,25,1598), (4,140,1656), (4,186,24), (4,376,20), (4,378,1593), (4,381,3843), (4,382,3830), (4,548,3534), (4,549,3526), (4,550,1657), (4,631,2), (4,632,5079), (4,1006,405), (4,1593,1597), (4,1656,3851), (4,1657,5073), (4,1658,2070), (4,2937,5899), (4,3088,1595), (4,3089,1596), (4,3090,3091), (4,3091,546), (4,3146,3627), (4,3147,3542), (4,3515,3517), (4,3520,378), (4,3522,550), (4,3523,140), (4,3524,631), (4,3525,3090), (4,3526,5055), (4,3528,376), (4,3529,3146), (4,3530,5054), (4,3533,5056), (4,3541,427), (4,3542,235), (4,3543,3853), (4,3545,3832), (4,3548,2072), (4,3627,5076), (4,3628,5072), (4,3832,3845), (4,3839,3861), (4,3855,3839), (4,5054,5070), (4,5056,3850), (4,5067,3545), (4,5068,3858), (4,5071,3855), (4,6353,3089), (4,6621,6624), (4,6622,6623), (5,20,382), (5,26,25), (5,140,2), (5,376,1657), (5,381,3851), (5,382,3843), (5,427,5576), (5,546,3091), (5,547,5056), (5,548,20), (5,549,140), (5,631,3526), (5,632,3628), (5,1656,5055), (5,1657,3830), (5,1658,24), (5,3090,5079), (5,3091,5072), (5,3522,3534), (5,3523,5054), (5,3526,5070), (5,3529,5076), (5,3530,631), (5,3534,5073), (5,3627,546), (5,3628,3090), (5,3845,3850), (5,3850,3545), (5,3853,3832), (5,3858,5066), (5,3861,3855), (5,5066,5068), (5,5498,6143), (5,6642,5020), (5,6644,6642), (6,182,5050), (6,187,1384), (6,371,3311), (6,372,3312), (6,574,5024), (6,1151,371), (6,1152,372), (6,1351,5093), (6,1620,1192), (6,2076,5017), (6,3053,32), (6,3311,6417), (6,3312,6418), (6,3592,6419), (6,3594,6420), (6,4252,58), (6,4255,386), (6,4258,4251), (6,5013,39), (6,5022,4253), (6,5023,3053), (6,5085,182), (6,5102,5097), (6,5210,187), (6,5585,5210), (6,6200,6221), (6,6221,6199), (6,6396,6398), (6,6398,6395), (6,6409,1151), (6,6410,1152), (6,6411,6200), (6,6412,6396), (6,6417,6500), (6,6418,6501), (6,6419,6427), (6,6420,6428), (6,6425,3592), (6,6426,3594), (6,6433,6437), (6,6434,6438), (6,6451,6445), (6,6452,6446), (6,6455,6407), (6,6456,6408), (7,3487,6147), (7,5703,3487), (8,100,5687), (8,2975,956), (8,5657,5690), (8,5731,944), (9,936,5044), (9,1490,5777), (9,5438,936), (10,355,5790), (10,993,958), (10,5267,993), (10,5745,5791), (11,5433,499), (11,6284,1479), (12,5432,498), (15,16,6), (15,62,61), (15,3364,371), (15,3365,372), (15,5237,62), (15,5352,5238), (16,61,62), (16,3389,371), (16,3390,372), (16,5238,61), (16,5351,5237), (20,21,1012), (20,140,381), (20,186,26), (20,376,550), (20,381,5073), (20,404,3149), (20,417,6638), (20,549,1656), (20,550,3534), (20,631,5), (20,1006,3560), (20,1656,3830), (20,1658,2937), (20,2060,3079), (20,3090,3627), (20,3091,3146), (20,3146,3529), (20,3522,376), (20,3523,2), (20,3524,140), (20,3525,546), (20,3526,3843), (20,3528,548), (20,3530,3526), (20,3533,3845), (20,3543,5059), (20,3628,5076), (20,5054,3851), (20,5056,3543), (20,5067,3853), (21,404,2), (21,411,4), (21,416,1982), (21,1816,29), (21,1817,28), (21,3658,3109), (21,4188,474), (21,4203,4195), (21,4225,859), (22,24,26), (22,26,2937), (22,381,5899), (22,426,6638), (22,631,6642), (22,1599,3155), (22,1600,3156), (22,1995,23), (22,6644,2070), (23,1995,25), (24,25,3517), (24,26,2070), (24,186,3515), (24,378,4), (24,1593,1598), (24,1657,5899), (24,3516,1597), (24,3520,1593), (25,378,1597), (25,426,6617), (25,1593,4), (25,3515,24), (25,3516,1593), (26,140,6642), (26,378,382), (26,382,5899), (26,6642,3517), (26,6644,24), (28,4219,4), (29,412,4), (32,39,6), (32,182,3398), (32,187,3053), (32,574,39), (32,3053,1384), (32,5171,2080), (32,5206,187), (33,1753,1872), (35,36,1), (35,56,3295), (35,5010,5217), (35,5204,999), (35,5563,3746), (35,5584,6244), (36,55,999), (36,165,3428), (36,2078,5126), (36,3746,5563), (36,5010,55), (36,5217,3295), (39,187,32), (39,574,5013), (39,5008,5041), (39,5013,5024), (39,5023,1384), (39,5206,3053), (40,57,5709), (40,165,3579), (40,1385,1482), (40,3576,1), (41,672,218), (46,3612,1), (48,71,219), (50,566,6), (52,389,568), (52,569,6), (55,56,1), (55,165,6244), (55,3303,3746), (55,3304,3303), (55,5204,56), (55,5217,35), (55,5584,40), (56,1466,57), (56,3303,3304), (56,3304,5563), (56,5204,36), (56,5217,55), (56,5584,3428), (57,942,5708), (57,1420,1467), (57,3601,1), (58,386,6), (58,580,5398), (58,4256,386), (58,4257,4252), (58,4276,4267), (58,4278,3286), (61,62,6), (61,5238,15), (61,5351,16), (61,5864,1351), (62,5237,16), (62,5352,15), (62,5865,1351), (63,72,3927), (63,78,72), (63,3984,3951), (63,4652,3916), (63,4855,78), (63,5440,3940), (64,154,1498), (65,1155,46), (65,2646,1), (69,3926,3933), (69,6337,3926), (71,1818,3781), (72,78,3940), (72,3916,63), (72,5440,78), (73,255,3157), (73,603,222), (74,1511,399), (74,1614,6241), (76,99,1975), (76,1078,183), (78,1259,1260), (78,3916,3927), (78,3951,3984), (78,4652,63), (78,4855,5440), (84,936,5777), (84,5044,5779), (84,5438,5720), (99,1078,76), (99,5152,5989), (100,2975,8), (100,5303,2975), (101,3730,220), (104,5657,956), (110,1614,156), (140,376,382), (140,381,5070), (140,382,5055), (140,546,3628), (140,549,631), (140,550,4), (140,631,5054), (140,632,3525), (140,1368,3548), (140,1657,3851), (140,1658,6644), (140,3146,5079), (140,3522,1657), (140,3528,3534), (140,3529,5072), (140,3530,549), (140,3534,3843), (140,3627,3090), (140,3628,632), (140,3845,5067), (140,3853,547), (140,5428,1006), (140,6636,2937), (143,5946,3567), (155,1147,3167), (157,160,159), (165,5010,2077), (165,6282,3587), (171,5329,1460), (182,576,575), (182,578,569), (182,1160,6418), (182,1161,6417), (182,1350,1351), (182,5092,5085), (182,5171,32), (183,1975,76), (184,185,1181), (184,394,3167), (184,1092,1147), (184,1147,49), (184,1204,185), (184,3917,394), (184,5562,155), (185,1092,155), (185,3917,5562), (186,376,22), (186,378,25), (186,550,2937), (186,1593,3517), (186,3516,1598), (186,3520,4), (186,3651,2915), (187,574,6), (187,2021,1691), (187,5162,2076), (187,5188,5171), (187,5206,5023), (191,6326,5694), (198,1436,610), (199,1011,25), (199,3145,2915), (212,603,255), (212,4303,3157), (216,577,6), (216,3284,5158), (220,3207,101), (230,5254,3767), (232,1968,2207), (235,468,3542), (235,1885,4), (237,3148,25), (243,1940,158), (255,4303,222), (283,1790,1437), (284,579,6), (371,372,6), (371,1151,6221), (371,1152,3312), (371,1350,1161), (371,2459,6423), (371,3103,6422), (371,3311,6199), (371,3312,6417), (371,3594,6427), (371,6200,1151), (371,6395,6500), (371,6396,372), (371,6398,6418), (371,6409,6449), (371,6410,6398), (371,6411,6455), (371,6412,6450), (371,6419,3592), (371,6420,6419), (371,6425,6447), (371,6426,6428), (371,6449,6407), (371,6450,6395), (371,6452,6408), (371,6453,6425), (371,6454,6420), (371,6455,6445), (371,6481,6432), (371,6484,6429), (371,6486,6480), (371,6497,6446), (372,1151,3311), (372,1152,6398), (372,1350,1160), (372,2460,6424), (372,3102,6421), (372,3311,6418), (372,3312,6395), (372,3592,6428), (372,6199,6501), (372,6200,371), (372,6221,6417), (372,6396,1152), (372,6409,6221), (372,6410,6450), (372,6411,6449), (372,6412,6456), (372,6419,6420), (372,6420,3594), (372,6425,6427), (372,6426,6448), (372,6449,6199), (372,6450,6408), (372,6451,6407), (372,6453,6419), (372,6454,6426), (372,6456,6446), (372,6480,6431), (372,6485,6430), (372,6487,6481), (372,6496,6445), (376,549,381), (376,631,4), (376,1006,1012), (376,3090,3529), (376,3522,548), (376,3523,5), (376,3524,2), (376,3525,3146), (376,3526,5073), (376,3528,3522), (376,3530,1656), (376,5054,3830), (376,5067,5059), (378,2070,3830), (378,2937,5073), (378,3515,1598), (378,3520,3516), (378,6644,381), (381,382,4), (381,1656,5), (381,1657,382), (381,2070,25), (381,3526,1656), (381,5054,2), (381,5072,3091), (381,5079,5072), (382,631,5070), (382,1656,381), (382,3526,5), (382,3534,1657), (382,5054,1656), (382,5076,3627), (382,5079,546), (384,3552,1003), (384,5999,4), (386,573,970), (386,581,5396), (386,991,581), (386,4256,4255), (386,4257,58), (386,5752,5754), (388,3085,495), (388,5218,3085), (389,578,6), (394,1181,155), (394,3796,184), (394,5406,5408), (394,5407,5409), (404,1006,140), (404,4189,405), (404,6636,2915), (405,474,2), (405,1012,3560), (405,2915,25), (405,3149,5), (408,4189,6638), (411,1006,5), (411,3523,474), (411,4189,1012), (417,1593,6617), (418,6641,25), (426,3148,441), (426,6641,2), (427,3575,4), (428,1907,4), (454,3548,6617), (465,466,2), (468,1885,235), (474,1012,5), (474,3560,1656), (485,5418,590), (485,6560,3070), (486,5420,615), (486,6561,3071), (487,488,69), (489,492,637), (490,491,638), (497,3086,496), (498,1478,12), (498,4299,1478), (499,1479,11), (499,4302,1479), (500,582,6), (500,5396,581), (546,549,3525), (546,550,3529), (546,632,3090), (546,3090,5072), (546,3091,381), (546,3146,5076), (546,3525,1656), (546,3529,382), (546,3627,4), (546,3628,5), (546,5079,3851), (547,3543,381), (547,3845,3545), (547,3850,5), (547,3853,3850), (547,5067,1656), (548,549,4), (548,550,376), (548,631,382), (548,632,3529), (548,3523,381), (548,3524,1656), (548,3530,5), (548,5054,5073), (549,550,5), (549,1657,5070), (549,3522,382), (549,3528,1657), (549,3530,3523), (549,3534,5055), (549,3627,632), (549,3853,3533), (549,6636,2070), (550,631,381), (550,632,3627), (550,1656,5073), (550,1658,22), (550,3523,1656), (550,3524,3526), (550,3525,5076), (550,3526,3830), (550,3530,2), (550,3628,3146), (550,3850,5059), (550,5054,3843), (550,5498,3153), (551,5493,4301), (567,568,6), (567,3581,568), (568,6243,52), (569,578,567), (570,571,6), (572,573,6), (572,3430,581), (573,579,5755), (573,581,5752), (574,5171,3095), (574,5206,32), (574,5210,1384), (575,576,6), (577,578,2055), (577,5158,3284), (579,991,5751), (579,5751,5753), (580,581,6), (580,3430,5752), (581,991,500), (582,5398,580), (583,584,6), (590,3070,485), (595,995,1191), (601,602,31), (615,3071,486), (616,628,634), (617,627,633), (620,626,3788), (627,633,298), (628,634,299), (631,1657,5055), (631,3090,3525), (631,3091,632), (631,3146,3628), (631,3523,549), (631,3524,3523), (631,3528,20), (631,3529,3090), (631,3534,3851), (631,3545,3533), (631,3651,3149), (631,5059,547), (631,6636,26), (631,6643,3548), (632,3091,1656), (632,3146,5072), (632,3525,3526), (632,3529,381), (632,3627,5), (632,3628,2), (632,5079,5070), (800,5065,6), (902,1201,3915), (910,1212,169), (936,1490,5720), (936,5732,1490), (936,5777,5780), (938,4313,3488), (940,5706,5707), (942,5709,2095), (943,3487,954), (944,5657,8), (946,1125,5886), (950,1210,5722), (950,3911,1210), (956,5687,8), (958,1376,10), (962,3616,5603), (965,5776,5778), (970,5396,5754), (980,5337,940), (997,1158,5887), (999,3295,1), (1006,3651,4), (1011,4191,2), (1012,3149,4), (1030,5096,5132), (1030,5124,6), (1038,1040,1), (1060,1062,1), (1074,1076,225), (1092,1181,3167), (1092,5562,394), (1106,1253,1496), (1125,5248,1001), (1147,1216,394), (1150,5767,5769), (1151,1152,6), (1151,3312,6199), (1151,3592,6425), (1151,3594,3592), (1151,6200,6449), (1151,6221,6407), (1151,6396,3312), (1151,6398,6417), (1151,6408,6500), (1151,6409,6200), (1151,6410,372), (1151,6411,6409), (1151,6412,1152), (1151,6419,6447), (1151,6425,6453), (1151,6426,6419), (1151,6429,6480), (1151,6431,6437), (1151,6433,6484), (1151,6437,6429), (1151,6438,6431), (1151,6446,6501), (1151,6449,6445), (1151,6450,6418), (1151,6453,6519), (1151,6454,6427), (1151,6456,6395), (1151,6497,6408), (1152,3311,6395), (1152,3592,3594), (1152,3594,6426), (1152,6200,3311), (1152,6221,6418), (1152,6396,6450), (1152,6398,6408), (1152,6407,6501), (1152,6409,371), (1152,6410,6396), (1152,6411,1151), (1152,6412,6410), (1152,6420,6448), (1152,6425,6420), (1152,6426,6454), (1152,6430,6481), (1152,6432,6438), (1152,6434,6485), (1152,6437,6432), (1152,6438,6430), (1152,6445,6500), (1152,6449,6417), (1152,6450,6446), (1152,6453,6428), (1152,6454,6522), (1152,6455,6199), (1152,6496,6407), (1155,2646,65), (1160,3311,1351), (1161,3312,1351), (1180,1627,5359), (1191,3052,595), (1193,4300,1064), (1210,4304,950), (1216,5447,3917), (1312,1313,2072), (1319,3057,1), (1333,4261,6), (1340,1341,182), (1340,1380,6), (1341,1379,6), (1342,1343,182), (1342,1670,6), (1343,1671,6), (1344,1345,381), (1350,5023,5171), (1350,5085,6), (1350,5092,5050), (1351,5050,6), (1368,3547,1656), (1384,5024,6), (1385,6585,1617), (1388,2098,1), (1399,2361,47), (1420,1697,1), (1428,3056,613), (1469,2330,611), (1470,5172,1617), (1479,4302,6284), (1495,5650,5651), (1504,1505,1570), (1504,5062,6), (1505,5058,6), (1532,4187,5), (1578,1579,6), (1583,1584,2), (1583,3156,5020), (1584,3155,5020), (1589,1590,2), (1593,3515,25), (1593,3516,378), (1593,6642,381), (1594,6240,4), (1597,1598,4), (1597,3517,1598), (1597,5020,381), (1598,3517,25), 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(6449,6497,6398), (6449,6522,6419), (6450,6451,371), (6450,6452,6456), (6450,6455,3311), (6450,6456,6396), (6450,6496,6221), (6450,6497,6410), (6450,6519,6420), (6451,6452,6), (6451,6455,6409), (6451,6456,3311), (6451,6497,3312), (6451,6522,6425), (6452,6455,3312), (6452,6456,6410), (6452,6496,3311), (6452,6519,6426), (6453,6454,6), (6453,6482,6480), (6453,6484,6482), (6453,6519,6407), (6454,6483,6481), (6454,6485,6483), (6454,6522,6408), (6455,6456,6), (6455,6496,6451), (6455,6497,372), (6456,6496,371), (6456,6497,6452), (6465,6466,6467), (6468,6469,6), (6468,6471,6470), (6469,6470,6471), (6470,6471,6), (6472,6473,6), (6472,6474,6221), (6473,6475,6398), (6474,6475,6), (6476,6477,6), (6478,6479,6), (6480,6481,6), (6480,6484,1151), (6480,6486,6484), (6480,6487,6432), (6481,6485,1152), (6481,6486,6431), (6481,6487,6485), (6482,6483,6), (6482,6487,6420), (6483,6486,6419), (6484,6485,6), (6484,6486,6433), (6485,6487,6434), (6486,6487,6), (6488,6489,6), (6490,6491,6), (6492,6493,6), (6494,6495,6), (6494,6499,6435), (6495,6498,6436), (6496,6497,6), (6496,6522,6519), (6497,6519,6522), (6498,6499,6), (6500,6501,6), (6519,6522,6), (6566,6567,1570), (6639,6640,2)

X(4) = ORTHOCENTER

Trilinears sec A : sec B : sec CTrilinears cos A - sin B sin C : cos B - sin C sin A : cos C - sin A sinBTrilinears cos A - cos(B - C) : cos B - cos(C - A) : cos C - cos(A - B)Trilinears sin B sin C - cos(B - C) : sin C sin A - cos(C - A) : sin A sin B - cos(A - B)Trilinears csc A tan 3A - 2 sec 3A : :Trilinears 4 cos A - cos(B - C) - 3 sin B sin C : :Trilinears cos A + cos(B - C) + 5 cos B cos C - 2 sin B sin C : :Barycentrics 1/SA : 1/SB : 1/SCBarycentrics tan A : tan B : tan CBarycentrics 1/(b+ c- a) : 1/(c+ a- b) : 1/(a+ b- cTripolars |cos A| : :Tripolars |a(b^2 + c^2 - a^2)| : :

As a point on the Euler line, X(4) has Shinagawa coefficients (0, 1).

X(4) is the point of concurrence of the altitudes of ABC.

The tangents at A,B,C to the McCay cubic K003 concur in X(4). Also, the tangents at A,B,C to the Lucas cubic K007 concur in X(4). (Randy Hutson, November 18, 2015)

Let P be a point in the plane of ABC. Let Oa be the circumcenter of BCP, and define Ob and Oc cyclically. Let Q be the circumcenter of OaObOc. P = Q only when P = X(4). (Randy Hutson, June 27, 2018)

Taking a reference triangle ABC and a variable point P on the plane, P=X(3) is the point of maximal area of its pedal triangle when considering all points P inside the circumcircle of ABC. There are points P far away from the circumcircle for which the area of their pedal triangles is much larger. However, if you consider the signed area of the pedal triangle of P (of which sign depends on whether the points are in clockwise or anti-clockwise order), you could just say that the area of the pedal triangle of P is always negative whenever P is outside of the circumcircle so that P=X(3) gives the global maximum. (Mark Helman, July 10, 2020)

A slightly similar thing happens regarding the area of the antipedal triangle of P. P=X(4) has the smallest area of its antipedal amongst all P in the interior of triangle ABC (when X(4) is in this interior). There are points P (on the circumcircle) for which this area goes to 0. However, if we consider the signed area of the antipedal, even though there are still regions of the plane outside of ABC where the signed area is positive, P=X(4) gives the smallest area of the antipedal among all P for which this area is positive (this works even when ABC is obtuse, and points close to the circumcircle (on both sides) have negative antipedal area). (Mark Helman, July 10, 2020)

View Extremal Area Pedal and Antipedal Triangles, by Mark Helman, Ronaldo Garcia, and Dan Reznik.

If you have The Geometer's Sketchpad, you can view Orthocenter.

If you have GeoGebra, you can view Orthocenter.

X(4) and the vertices A,B,C comprise an orthocentric system, defined as a set of four points, one of which is the orthocenter of the triangle of the other three (so that each is the orthocenter of the other three). The incenter and excenters also form an orthocentric system. If ABC is acute, then X(4) is the incenter of the orthic triangle.

Suppose P is not on a sideline of ABC. Let A'B'C' be the cevian triangle of P. Let D,E,F be the circles having diameters AA', BB',CC'. The radical center of D,E,F is X(4). (Floor van Lamoen, Hyacinthos #214, Jan. 24, 2000.)

Let A2B2C2 be the 2nd Conway triangle. Let A' be the crosspoint of B2 and C2, and define B' and C' cyclically. The lines AA', BB', CC' concur in X(4). (Randy Hutson, December 10, 2016)

Let T be any one of these trianges: {anticevian of X(30), anti-Hutson-intouch, anti-incircle-circles, Ehrmann side, X(2)-Ehrmann, Gemini 15, Gemini 16, Kosnita, midheight, N-obverse of X(6), Schroeter, tangential, Trinh, 1st Zaniah, 2nd Zaniah}. Let O A be the circle centered at the A-vertex of T and passing through A; define O B and O C cyclically. X(4) is the radical center of O A , O B , O C . (Randy Hutson, August 30, 2020)

See Ross Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, Mathematical Association of America, 1995. Chapter 2: The Orthocenter.

X(4) lies on the Thomson, Darboux, Napoleon, Lucas, McCay, and Neuberg cubics, and the Darboux septic, and on these lines: {1,33}, {2,3}, {6,53}, {7,273}, {8,72}, {9,10}, {11,56}, {12,55}, {13,61}, {14,62}, {15,17}, {16,18}, {31,3072}, {32,98}, {35,498}, {36,499}, {37,1841}, {39,232}, {41,2202}, {42,1860}, {46,90}, {48,1881}, {49,156}, {50,9220}, {51,185}, {52,68}, {54,184}, {57,84}, {58,5292}, {63,5709}, {64,459}, {65,158}, {66,9969}, {67,338}, {69,76}, {74,107}, {75,12689}, {78,908}, {79,1784}, {80,1825}, {81,5707}, {83,182}, {85,4872}, {93,562}, {94,143}, {95,8797}, {96,231}, {99,114}, {100,119}, {101,118}, {102,124}, {103,116}, {105,5511}, {106,5510}, {109,117}, {110,113}, {111,1560}, {120,1292}, {121,1293}, {122,1294}, {123,1295}, {126,1296}, {127,1289}, {128,930}, {129,1303}, {130,1298}, {131,135}, {137,933}, {141,1350}, {142,5732}, {144,2894}, {145,149}, {147,148}, {150,152}, {151,2818}, {154,8888}, {155,254}, {157,5593}, {160,3613}, {162,270}, {165,1698}, {171,601}, {175,10905}, {176,10904}, {177,8095}, {181,9553}, {183,3785}, {187,7607}, {189,5908}, {191,2949}, {193,1351}, {195,399}, {200,6769}, {201,7069}, {204,1453}, {210,7957}, {212,3074}, {214,12119}, {215,9652}, {216,8799}, {218,294}, {230,3053}, {233,10979}, {236,8128}, {238,602}, {240,256}, {250,1553}, {251,8879}, {252,1487}, {255,1935}, {276,327}, {279,1565}, {280,2968}, {282,3345}, {284,5747}, {290,6528}, {298,5864}, {299,5865}, {312,7270}, {325,1975}, {333,5788}, {339,10749}, {341,12397}, {345,7283}, {346,3695}, {347,6356}, {348,5088}, {354,3296}, {371,485}, {372,486}, {373,11465}, {385,7823}, {386,2051}, {390,495}, {391,2322}, {394,1217}, {477,1304}, {484,3460}, {487,489}, {488,490}, {493,8212}, {494,8213}, {496,999}, {512,879}, {518,6601}, {519,3680}, {523,1552}, {524,5485}, {525,2435}, {527,5735}, {528,3913}, {529,3813}, {532,5862}, {533,5863}, {535,8666}, {538,7758}, {539,9936}, {541,9140}, {542,576}, {543,5503}, {544,10710}, {551,9624}, {566,9221}, {567,7578}, {569,1179}, {572,1474}, {574,1506}, {575,598}, {579,1713}, {580,1714}, {584,8818}, {590,1151}, {595,8750}, {603,3075}, {604,7120}, {608,1518}, {615,1152}, {616,627}, {617,628}, {618,5473}, {619,5474}, {620,7862}, {625,3788}, {626,3734}, {635,3643}, {636,3642}, {639,5590}, {640,5591}, {641,12124}, {642,12123}, {651,3157}, {653,1156}, {674,12587}, {684,2797}, {685,2682}, {690,11005}, {693,8760}, {695,3981}, {754,7751}, {758,5693}, {774,1254}, {800,13380}, {801,1092}, {842,935}, {885,3309}, {912,3868}, {916,2997}, {936,3452}, {937,1534}, {940,1396}, {941,1880}, {953,1309}, {958,2886}, {960,5794}, {970,9534}, {972,5514}, {973,6145}, {974,7729}, {983,5255}, {990,4000}, {991,4648}, {993,11012}, {1000,3057}, {1015,9651}, {1029,2906}, {1032,5910}, {1034,5911}, {1036,1065}, {1037,1067}, {1038,1076}, {1039,1096}, {1040,1074}, {1041,2263}, {1043,4417}, {1046,2648}, {1060,4296}, {1062,3100}, {1073,2130}, {1078,5171}, {1089,3974}, {1104,3772}, {1111,4056}, {1123,7133}, {1125,3576}, {1131,3311}, {1132,3312}, {1138,2132}, {1139,3368}, {1140,3395}, {1157,2120}, {1160,1162}, {1161,1163}, {1164,3595}, {1165,3593}, {1175,5320}, {1177,5622}, {1192,3532}, {1209,4549}, {1216,2979}, {1248,2660}, {1251,1832}, {1260,5687}, {1317,12763}, {1319,7704}, {1327,6419}, {1328,6420}, {1329,1376}, {1336,2362}, {1340,1348}, {1341,1349}, {1342,1676}, {1343,1677}, {1353,5093}, {1379,2040}, {1380,2039}, {1383,8791}, {1384,8778}, {1385,1538}, {1389,2099}, {1392,3241}, {1393,7004}, {1399,5348}, {1420,4311}, {1430,1468}, {1435,3333}, {1440,7053}, {1441,4329}, {1445,3358}, {1448,7365}, {1469,12589}, {1483,3623}, {1484,12773}, {1495,3431}, {1499,1550}, {1500,9650}, {1510,10412}, {1511,12121}, {1521,7115}, {1562,6529}, {1566,2724}, {1609,9722}, {1621,10267}, {1670,5404}, {1671,5403}, {1682,9552}, {1689,2010}, {1690,2009}, {1691,3406}, {1697,7160}, {1715,1730}, {1716,1721}, {1717,1718}, {1726,1782}, {1729,8558}, {1764,10479}, {1768,3065}, {1773,2961}, {1781,2955}, {1798,13323}, {1840,4876}, {1903,2262}, {1942,2790}, {1957,5247}, {1970,1971}, {1973,2201}, {1987,3269}, {1994,2904}, {2077,3814}, {2080,7793}, {2092,3597}, {2093,4848}, {2095,9965}, {2098,10944}, {2121,3481}, {2131,3349}, {2133,8440}, {2181,4642}, {2217,3417}, {2275,9597}, {2276,9596}, {2278,5397}, {2287,5778}, {2331,3755}, {2332,4251}, {2353,3425}, {2355,3579}, {2361,7299}, {2393,5486}, {2456,10349}, {2457,3667}, {2477,9653}, {2482,12117}, {2536,2540}, {2537,2541}, {2574,2592}, {2575,2593}, {2646,4305}, {2651,2907}, {2679,2698}, {2687,2766}, {2697,10423}, {2734,10017}, {2752,10101}, {2770,10098}, {2771,9803}, {2778,10693}, {2783,10769}, {2784,11599}, {2787,10768}, {2791,4516}, {2793,9180}, {2801,3254}, {2802,12641}, {2814,3762}, {2817,13532}, {2822,4466}, {2823,4858}, {2826,10773}, {2827,10774}, {2828,10775}, {2830,10779}, {2831,10780}, {2840,4939}, {2889,6101}, {2896,6194}, {2900,3189}, {2905,6625}, {2908,7139}, {2917,8146}, {2929,2935}, {2972,10745}, {2975,5841}, {2995,8048}, {3023,12185}, {3024,12374}, {3027,12184}, {3028,12373}, {3054,5210}, {3056,12588}, {3058,3303}, {3062,3339}, {3094,3399}, {3096,3098}, {3101,8251}, {3120,3924}, {3162,5359}, {3164,9290}, {3172,3424}, {3180,5873}, {3181,5872}, {3184,6716}, {3190,3191}, {3212,7261}, {3216,5400}, {3218,5770}, {3255,5883}, {3270,11461}, {3304,5434}, {3305,3587}, {3306,7171}, {3314,7885}, {3320,12945}, {3329,7864}, {3338,7284}, {3340,3577}, {3342,3347}, {3344,3348}, {3352,3354}, {3356,3637}, {3364,3391}, {3365,3392}, {3366,3389}, {3367,3390}, {3369,3397}, {3370,3396}, {3371,3387}, {3372,3388}, {3373,3385}, {3374,3386}, {3379,5402}, {3380,5401}, {3381,3394}, {3382,3393}, {3398,3407}, {3413,3558}, {3414,3557}, {3416,3714}, {3426,13093}, {3430,3454}, {3438,3443}, {3439,3442}, {3440,5682}, {3441,5681}, {3461,7165}, {3463,5683}, {3466,3469}, {3479,3489}, {3480,3490}, {3495,8866}, {3497,7351}, {3499,8925}, {3500,7350}, {3502,8867}, {3521,5946}, {3527,8796}, {3580,11472}, {3582,4325}, {3584,4330}, {3589,5085}, {3590,6221}, {3591,6398}, {3601,4304}, {3611,11460}, {3614,5217}, {3617,5690}, {3620,7879}, {3621,5844}, {3622,5901}, {3624,7987}, {3629,5102}, {3632,4900}, {3633,11224}, {3634,10164}, {3648,3652}, {3668,8809}, {3671,5665}, {3679,4866}, {3701,5300}, {3704,5695}, {3706,10371}, {3738,10771}, {3741,10476}, {3746,4309}, {3753,9800}, {3812,5880}, {3815,5013}, {3819,13348}, {3820,6244}, {3822,5248}, {3825,10200}, {3826,11495}, {3829,11194}, {3841,7688}, {3847,6691}, {3849,7615}, {3870,5534}, {3871,10528}, {3877,7700}, {3885,12648}, {3887,10772}, {3911,6705}, {3916,5744}, {3917,7999}, {3925,5584}, {3933,7776}, {3934,5188}, {3940,5763}, {3947,4314}, {3972,7828}, {4008,12723}, {4045,7808}, {4048,5103}, {4277,4646}, {4308,7743}, {4313,5226}, {4316,7280}, {4317,5563}, {4324,5010}, {4339,5266}, {4355,10980}, {4357,10444}, {4423,7958}, {4425,8235}, {4444,6002}, {4512,10268}, {4645,7155}, {4654,11518}, {4658,5733}, {4692,4894}, {4721,4805}, {4723,12693}, {4768,9525}, {4846,5462}, {4847,12527}, {4863,12692}, {5007,5309}, {5008,5346}, {5032,11405}, {5038,11170}, {5044,10157}, {5045,5558}, {5050,5395}, {5092,7859}, {5097,7894}, {5119,7162}, {5121,11512}, {5123,13528}, {5173,12677}, {5204,5433}, {5206,6781}, {5221,10308}, {5223,12777}, {5249,10884}, {5253,10269}, {5265,10593}, {5273,5791}, {5278,9958}, {5281,10592}, {5377,6074}, {5418,6200}, {5420,6396}, {5424,5441}, {5435,5704}, {5437,9841}, {5438,6700}, {5439,9776}, {5440,5748}, {5447,7998}, {5449,7689}, {5461,10153}, {5505,10752}, {5513,9085}, {5533,10074}, {5535,6597}, {5536,6763}, {5542,6744}, {5550,11230}, {5553,7702}, {5556,10977}, {5557,12005}, {5559,5697}, {5561,11552}, {5597,8196}, {5598,8203}, {5599,11822}, {5600,11823}, {5601,8200}, {5602,8207}, {5606,5950}, {5609,5655}, {5623,8446}, {5624,8456}, {5627,6070}, {5670,8487}, {5671,8494}, {5672,8444}, {5673,8454}, {5674,8495}, {5675,8496}, {5676,8486}, {5677,7329}, {5678,8491}, {5679,8492}, {5680,7164}, {5685,8480}, {5688,12698}, {5689,12697}, {5705