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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 $\sum_{i=1}^n(-1)^it_i=0$.

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    Sorry but drawing is too difficult for me as I have a cerebral palsy.2012-05-11
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    "the length of the tangent from $P$ to the circle tangent to $C$ at $v_i$ But there are two possible tangents, no?2012-05-11
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    @leonbloy: But they both have the same length.2012-05-11
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    The description seems pretty straightforward to me. But I have no idea how to prove the thing.2012-05-11
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    @TonyK: of course, silly me :-)2012-05-11
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    The theorem was stated of the book "Which Way did the Bicycle Go" and it was said that this is a generalization of the Casey's theorem. Furthermore, the book says the proof can be found on the book "Ptolemy's Legacy" but I don't have that book.2012-05-11

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