Problem #PRU-5081

Problem

Theorem: If we mark $n$ points on a circle and connect each point to every other point by a straight line, the lines divide the interior of the circle is into is $2n-1$ regions.
"Proof": First, let’s have a look at the smallest natural numbers.

• 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 $2n-1$ 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 $2\times (2n-1)=2n$ regions.