RO-TST 2010 Problem 21

Let $ABC$ be a scalene triangle, let $I$ be its incentre, and let $A_1$ , $B_1$ and $C_1$ be the points of contact of the excircles with the sides $BC$ , $CA$ and $AB$ , respectively. Prove that the circumcircles of the triangles $AIA_1$ , $BIB_1$ and $CIC_1$ have a common point different from $I$ .

Cezar Lupu & Vlad Matei