MathNet 2023-part-04 — 191 problems (EN–VI) Question 27

Let $M$ be the midpoint of side $AC$ of triangle $ABC$. Let $P$ be a point inside triangle $ABC$ such that $\angle BAP = \angle BCP$. The line $CP$ intersects side $AB$ at point $R$. Let $Q$ be the foot of the perpendicular drawn from $B$ to line $CP$. If $Q$ lies inside triangle $ABC$ and $QM = \frac{PC}{2}$ then prove that $RQ = QP$.
(Khulan Tumenbayar)