Question 99
The remote is $AB$. At this distance, we choose any point $P$, different from $A$ and $B$, and construct isosceles right triangles $APQ$ and $PBR$, whose vertices $Q$ and $R$ lie on the same side of the line through $A$ and $B$. Let the point $M$ be the halfway point of the distance $QR$. Prove that the distance of point $M$ from the line through $A$ and $B$ does not depend on the choice of point $P$.