KöMaL 2021 B5177

In a right-angled triangle $ABC$, line segment $CD$ is the altitude drawn to the hypotenuse. The circle $k$ of diameter $CD$ intersects the legs $AC$ and $BC$ again at points $E$ and $F$, respectively. The tangent drawn to circle $k$ at point $E$ intersects the line of leg $BC$ at point $P$, and the hypotenuse $AB$ at point $M$. The tangent drawn to circle $k$ at point $F$ intersects the line of leg $AC$ at point $Q$, and the hypotenuse $AB$ at point $N$. Prove that $4\cdot MN^2=PE^2+QF^2+2\cdot EF^2.$