Track / Stage 6 / 37 of 400 #1037 of 2000

Problem 1037

National olympiad, first round

Geometry Difficulty 6.0 Prove it

313. Consider a circle in which a regular $(2 n+1)$ -gon $A_{1} A_{2} \ldots A_{2 n+1}$ is inscribed. Let $A$ be an arbitrary point on the arc $A_{1} A_{2 n+1}$ .
a) Prove that the sum of the distances from $A$ to the vertices with even indices is equal to the sum of the distances from $A$ to the vertices with odd indices.

b) Construct equal circles that are tangent to the given circle at points $A_{1}, A_{2}, \ldots, A_{2 n+1}$ . Prove that the sum of the tangents drawn from $A$ to the circles tangent to the given circle at the vertices with even indices is equal to the sum of the tangents drawn to the circles tangent to the given circle at the vertices with odd indices.

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

313. a) Let $A$ be an arbitrary point on the circle ( $A$ is on the arc $A_{1} A_{2 n+1}$ ). Denote the side of the polygon by $a$ , and the length of the diagonal connecting vertices through one, by $b$ . By Ptolemy's theorem (problem II.237) for the quadrilateral $A A_{k} A_{k+1} A_{k+2}$ we have: $\left|A A_{k}\right| a + \left|A A_{k+2}\right| a = \left|A A_{k+1}\right| b (k=1,2, \ldots, 2 n-1)$ . Similar relations can be written for the quadrilaterals $A_{2 n} A_{2 n+1} A A_{1}$ and $A_{2 n+1} A A_{1} A_{2}:$

$\begin{gathered} \left|A A_{1}\right| a + \left|A A_{2 n+1}\right| b = \left|A A_{2 n}\right| a \\ \left|A A_{2 n+1}\right| a + \left|A A_{1}\right| b = \left|A A_{2}\right| a \end{gathered}$

By adding all these equations, leaving the vertices with even numbers on the right and the odd ones on the left, we obtain the required statement.
b) Our statement follows from part a) and the result of problem I. 206. (A similar formula can be obtained in the case of internal tangency.)

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.