IR-TST 2008 Problem 9

$I_a$ is the excenter of the triangle $ABC$ with respect to $A$ , and $AI_a$ intersects the circumcircle of $ABC$ at $T$ . Let $X$ be a point on $TI_a$ such that $XI_a^2=XA.XT$ . Draw a perpendicular line from $X$ to $BC$ so that it intersects $BC$ in $A'$ . Define $B'$ and $C'$ in the same way. Prove that $AA'$ , $BB'$ and $CC'$ are concurrent.