Question 177
Let $ABC$ be an acute triangle, and let $P$ be a point inside triangle $ABC$. Let $D$ be the midpoint of segment $[PC]$, $E$ the intersection point of lines $(AP)$ and $(BC)$, and $Q$ the intersection point of lines $(BP)$ and $(DE)$. Show that if the angles $\widehat{PAC}$ and $\widehat{PCB}$ are equal, then $\sin (\widehat{BCQ}) = \sin (\widehat{BAP})$.