Question 62
Let $ABC$ be a triangle on the plane. The angle bisector from the vertex $A$ meets the side $BC$ at $P$, and the median from the vertex $B$ meets the side $AC$ at $M$. The lines $AB$ and $MP$ meet at the point $K$. Prove that if $\frac{|PC|}{|BP|} = 2$, then $AP$ and $CK$ are perpendicular.