Question 47
In triangle $A B C$, let $I_{a}, I_{b}, I_{c}$ be the centers of the excircles tangent to sides $B C, C A, A B$, respectively. Let $P$ and $Q$ be the tangency points of the excircle of center $I_{a}$ with lines $A B$ and $A C$. Line $P Q$ intersects $I_{a} B$ and $I_{a} C$ at $D$ and $E$. Let $A_{1}$ be the intersection of $D C$ and $B E$. In an analogous way we define points $B_{1}$ and $C_{1}$. Prove that $A A_{1}, B B_{1}, C C_{1}$ are concurrent.