Question 156
Let $ABC$ be an acute triangle with $AT$, $AS$ respectively the internal and external angle bisectors of angle $A$ of triangle $ABC$, with $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle $ABC$. Denote by $D, E, F, I$ the incenters of triangles $PBC$, $PCA$, $PAB$, $ABC$ respectively. Prove that the four lines $AD$, $BE$, $CF$ and $IP$ are concurrent.