APMO 2001 Problem 3

Two equal-sized regular $n$ -gons intersect to form a $2n$ -gon $C$ . Prove that the sum of the sides of $C$ which form part of one $n$ -gon equals half the perimeter of $C$ .

Alternative formulation:

Let two equal regular $n$ -gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$ -gon (with $n\ge 3$ ). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue.

Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.