Quadrilateral $APBQ$.

Question

Quadrilateral $APBQ$ is incsribed in a circle $\omega$ with $\angle P = \angle Q = 90^{\circ}$ and $AP = AQ < BP$. Let $X$ be a variable point on segment $\overline{PQ}$. Line $AX$ meets $\omega$ again at $S$ (other than $A$). Point $T$ lies on arc $AQB$ of $\omega$ such that $\overline{XT}$ is perpendicular to $\overline{AX}$. Let $M$ denote the midpoint of chord $\overline{ST}$. As $X$ varies on segment $\overline{PQ}$, show that $M$ moves along a circle.

Answer 1

Here's a cleaner version of my attempt at an answer. It's still not complete but it's free of clutter and also identifies the circle $M$ is supposed to lie on. I also tried a solution based entirely on analytical geometry and I got very far but it's very messy. The picture below is a screen shot of an animation based on my analytical treatment. Too bad I can't upload the actual video since it's kinda cool to see $M$ moving in an arc of a circle as $X$ moves up and down.

Anyway, I am convinced that there is a relatively simple argument to prove that $\overline{CM}$ (see last picture below) is independent of $\mu$ but I haven't found it yet.

($|\mu| \le 1$ is a parameter that controls the movement of $X$ along the segment $\overline{PQ}$).