Problem

Source: Czech-Polish-Slovak Junior Match 2014, Team p4 CPSJ

Tags: area of a triangle, geometry, perpendicular, circle, areas, equal areas



Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Circle with center $M$ passing through point $ C$, intersects lines $AC ,BC$ for the second time at points $P,Q$ respectively. Point $R$ lies on segment $AB$ such that the triangles $APR$ and $BQR$ have equal areas. Prove that lines $PQ$ and $CR$ are perpendicular.