US-TST 2003 Problem 6

Let $\overline{AH_1}, \overline{BH_2}$ , and $\overline{CH_3}$ be the altitudes of an acute scalene triangle $ABC$ . The incircle of triangle $ABC$ is tangent to $\overline{BC}, \overline{CA},$ and $\overline{AB}$ at $T_1, T_2,$ and $T_3$ , respectively. For $k = 1, 2, 3$ , let $P_i$ be the point on line $H_iH_{i+1}$ (where $H_4 = H_1$ ) such that $H_iT_iP_i$ is an acute isosceles triangle with $H_iT_i = H_iP_i$ . Prove that the circumcircles of triangles $T_1P_1T_2$ , $T_2P_2T_3$ , $T_3P_3T_1$ pass through a common point.