Question 178
Let $I_{a}$ be the excenter of triangle $A B C$ with respect to $A$. The line $A I_{a}$ intersects the circumcircle of triangle $A B C$ at $T$. Let $X$ be a point on segment $T I_{a}$ such that $X I_{a}^{2}=X A \cdot X T$. The perpendicular line from $X$ to $B C$ intersects $B C$ at $A^{\prime}$. Define $B^{\prime}$ and $C^{\prime}$ in the same way. Prove that $A A^{\prime}$, $B B^{\prime}$ and $C C^{\prime}$ are concurrent.