# How to apply the Poincaré formula to a regular n-gon?

I've been trying to solve the following home task:

Choose $n$ points ($n\ge 2$) on the circle's circumference and connect them all with each other using chords. In result, the circle is divided into $R$ number of regions. Let $P$ be the number of intersection points and $S$ the number of (chord) segments.

Prove that $P-S+R=1$.

A hint is provided that it's not advised nor needed to try and find the exact counts for $P$, $S$ or $R$.

My first insight is that we can always create a regular polygon with $n$ sides if we put the points on circle equally spaced. So for example a pentagon has $P=5$, $S=20$ and $R=16$.

But here I'm stuck badly. Maybe we could apply Euler's polyhedron formula but I don't see how since it's for polyhedrons. The hint makes me think about an inductive approach. But if we start with $n$ points and add one more (that is, go from $n$-gon to ($n+1$)-gon), I don't see how we can make use of the induction assumption to proof the $(n+1)$ case.

Any insight is welcome!

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You may use Euler's formula: Imagine that your drawing is placed on the northern half of a sphere and the southern half is an "empty" $n$-gon. – Christian Blatter May 2 '12 at 15:40

To make it more clear what I meant to say: AFAIK Euler's formula is for planar graphs. Removal of the non-intersecting chords is trivial as you point out, but I fail to see how we can systematically remove an intersecting chord, since that has a variable effect on the $R$, $S$ and $P$. – randomguy May 2 '12 at 23:25