Question 74
Let $M$ be the midpoint of the hypotenuse $AB$ of the right triangle $ABC$. Point $P$ is chosen on the cathetus $CB$ so that $CP : PB = 1 : 2$. The straight line passing through $B$ meets the segments $AC, AP,$ and $PM$ at points $X, Y, \text{ and } Z$, respectively.
Prove that the bisector of the angle $PZY$ passes through point $C$ if and only if the bisector of the angle $PYX$ also passes through $C$.
Prove that the bisector of the angle $PZY$ passes through point $C$ if and only if the bisector of the angle $PYX$ also passes through $C$.