Implications of the Buried Treasure problem

Question

I was thinking about the Buried Treasure problem

From "1,2,3 infinity" page 35.

https://archive.org/details/OneTwoThreeInfinity_158

One of the implications would be:

Given:

$\triangle CAE$ and $\triangle EBD$ are isosceles right triangles $M$ is the midpoint of $CD$

Prove:

$\triangle AMB$ is an isosceles right triangle.

But, I haven't been able to work out a proof. At least, not one based solely on classical geometry.

Answer 1

$MQ$ is the perpendicular bisector of a $AB$ and this solution is solely based on classical geometry: