Question 31
Let $ABC$ be a triangle inscribed in a circle $(\omega)$ and $I$ is the incenter. Denote $D, E$ as the intersection of $AI, BI$ with $(\omega)$. And $DE$ cuts $AC, BC$ at $F, G$ respectively. Let $P$ be a point such that $PF \parallel AD$ and $PG \parallel BE$. Suppose that the tangent lines of $(\omega)$ at $A, B$ meet at $K$. Prove that three lines $AE, BD, KP$ are concurrent or parallel.