TUYMAADA 2005 Problem 4

In a triangle $ABC$ , let $A_{1}$ , $B_{1}$ , $C_{1}$ be the points where the excircles touch the sides $BC$ , $CA$ and $AB$ respectively. Prove that $A A_{1}$ , $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.

Proposed by L. Emelyanov