Problem

Source: 2017 Saudi Arabia BMO TST I p2

Tags: geometry, incenter, concurrent



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