Problem

Source:

Tags: geometry, square, perpendicular, isosceles, right triangle



Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.