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Theorem: If we mark $n$ points on a circle and connect each point to every other point by a straight line, the number of regions that the interior of the circle is divided into is $2^{n-1}$.

Proof: First let's collect numerical evidence.

When $n = 1$ there is one region (the whole disc).

When $n = 2$ there are two regions (two half-discs).

When $n = 3$ there are 4 regions (three lune-like regions and one triangle in the middle).

When $n = 4$ there are 8 regions, and if you're still not convinced then try $n=5$ and you'll find 16 regions if you count carefully.

Our proof in general will be by induction on $n$. Assuming the theorem is true for $n$ points, consider a circle with $n+1$ points on it. Connecting $n$ of them together in pairs produces $2^{n-1}$ regions in the disc, and then connecting the remaining point to all the others will divide the previous regions into two parts, thereby giving us $2 \cdot 2^{n-1} = 2^n$ regions.

Or does it...