OBM 2012 Problem 2

$ABC$ is a non-isosceles triangle. $T_A$ is the tangency point of incircle of $ABC$ in the side $BC$ (define $T_B$ , $T_C$ analogously). $I_A$ is the ex-center relative to the side BC (define $I_B$ , $I_C$ analogously). $X_A$ is the mid-point of $I_BI_C$ (define $X_B$ , $X_C$ analogously).

Show that $X_AT_A$ , $X_BT_B$ , $X_CT_C$ meet in a common point, colinear with the incenter and circumcenter of $ABC$ .