DK-GEORGMOHR 2005 Problem 3

The point $P$ lies inside $\vartriangle ABC$ so that $\vartriangle BPC$ is isosceles, and angle $P$ is a right angle. Furthermore both $\vartriangle BAN$ and $\vartriangle CAM$ are isosceles with a right angle at $A$ , and both are outside $\vartriangle ABC$ . Show that $\vartriangle MNP$ is isosceles and right-angled.