Question 16
On the sides $AB$ and $AC$ of triangle $ABC$, there are points $P$ and $Q$ respectively such that $\angle APC = \angle AQB = 45^\circ$. Let the perpendicular line to side $AB$ through $P$ intersects line $BQ$ at $S$. Let the perpendicular line to side $AC$ through $Q$ intersects line $CP$ at $R$. Let $D$ be on side $BC$ such that $AD \perp BC$. Prove that the lines $PS, AD, QR$ meet at a common point and lines $SR$ and $BC$ are parallel.