MATHEMATICAL NOTES

EDITED BY E. F. BECKENBACIE,

University

of California

Material for this department should be sent directly to E. F. Beckenbach, University of

California,

Los

Angeles

24,

California.

NEW PROOF OF A MINIMUM PROPERTY OF THE REGULAR n-GON

L.

F. ToTH, Budapest, Hungary

J. Kurschik gives in his paper

Ober

dem Kreis

ein-

und urmgeschriebene

Vielecke*

among others

a

complete and entirely elementary geometrical proof

of the well known fact according to which the regular n-gon Pn has

a

minimal

area among all n-gons

P

circumscribed about a circle c. In this proof

P.

is

carried,

after

a

dismemberment

and

a suitable reassembly,

in n

-I

steps

into

Pn

so

that

the

area increases at

every

step.

In

this not e we give an extremely simple proof,t which appears

to

be new,

showing imm ediately that if Pn is not regular, then P,

,,,

where

the area

is

denoted by the same symbol as the domain.

Consider the circle C circumscr ibed about P. We

show

that already

for

the

part Pn- C of

P.

lying in C we have

Pn-C

>

Pn.

We

have Pn

C=C-ns+(S1S2+S2s3+

**

+SnSI),

where

we

denote

by

S, S2,

S* *

,

sn

the circular sect ions of C cut off by the consecutive sides of

Pn,

and

by s the circular section of C cut off by a tangent to c. Hence

Pn*C

_ C - nS.

Then

P7 >!

Pn

C

>

C

-

ns

=Pn.

Equality holds

in

P.

r!

P.C

resp. in

PnC

?

Pn

only if no vertex

of

Pn

lies

in

the

outside

resp.

in

the

inside

of

C;

this

completes

the

proof.

BINOMIAL

COEFFICIENTS MODULO A PRIME

N. J. FINE,

University of

Pennsylvania

The

following

theorem,

although

given by

Lucas

in

his

Theorie

des

Nombres

(pp.

417-420),

does not

appear

to be as

widely

known as

it

deserves

to

be:

THEOREM 1.

Let p be

a prime,

and let

M

=

MO

+

MIP

+

M2p2

+

*

.

+

Mkpk

(O

<

Mr

<

P),

No

+ Np + N2p2+

**

+Nkpk

(O

5

Nr

<

P).

*

Mathematische Annalen 30 (1887), pp. 578-581.

t

As

P. Sz6sz

remarked [Bemerkung zu einer

Arbeit von K.

Ktirschik,

Matematikai

es

Fizikai Lapok

XLIV (1937), p. 167, note 3]

Kurschik's

proof is independent of the axiom

of

parallels.

This

advantage

is

preserved in the present

proof.

589

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