Track / Stage 5 / 160 of 400 #760 of 2000

Problem 760

AIME late

Geometry Difficulty 5.4 Prove it

## Task 19/69

Given is an arbitrary n-gon that has two axes of symmetry $s_{1}$ and $s_{2}$ . Prove that $s_{1}$ and $s_{2}$ intersect inside the n-gon!

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

Assume that $s_{1}$ and $s_{2}$ do not coincide and do not intersect inside the $n$ -gon. Then they divide the area of the $n$ -gon into three regions $A_{1}, A_{2}$ , and $A_{3}$ . Due to the properties of symmetry, it holds that: $A_{1}=A_{2}+A_{3}, A_{3}=A_{1}+A_{2}$ .

From this, it follows that $A_{1}=A_{2}+A_{1}+A_{2}$ , thus $A_{2}=0$ . This means, however, that $s_{1}$ and $s_{2}$ coincide, which contradicts the assumption.

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