Some Lemmas

Lemma 1. If isosceles triangles $ZMX$ and $YMW$ have right angles at $M$, then $YX$ and $ZW$ are perpendicular and congruent. See Figure 1.
Lemma 2. If $Z$ and $X$ are the centers of squares that lie on sides of $ABC$, built towards the exterior of $ABC$, and if $M$ is the midpoint of the third side, then $ZMX$ is isosceles and has a right angle at $M$. See Figure 2.