I found the following theorem in a book of mine without a proof. Could someone show me a proof of it?
Given a regular $n$-gon, with $n$ odd and vertices $v_1,\ldots,v_n$, and $C$ its circumcircle. At each $v_i$ draw a circle that is internally tangent to $C$ at $v_i$, and suppose all these tangent circles are congruent. Let $P$ be any point on the minor arc from $v_1$ to $v_n$ and let $t_i$ be the length of the tangent from $P$ to the circle tangent to $C$ at $v_i$. Then $\sum_{i=1}^n(-1)^it_i=0$.