IMO 2000 Problem 6

Let $AH_1, BH_2, CH_3$ be the altitudes of an acute angled triangle $ABC$ . Its incircle touches the sides $BC, AC$ and $AB$ at $T_1, T_2$ and $T_3$ respectively. Consider the symmetric images of the lines $H_1H_2, H_2H_3$ and $H_3H_1$ with respect to the lines $T_1T_2, T_2T_3$ and $T_3T_1$ . Prove that these images form a triangle whose vertices lie on the incircle of $ABC$ .