The shooting lemma
Problem
A circle $\omega$ touches a chord $AB$ of a circle $\Omega$ at $K$ and is internally tangent to $\Omega$ at $T$. Let $M$ be the midpoint of the arc $AB$ of $\Omega$ not containing $T$. Prove that the ray $TK$ passes through $M$, and that $MA^2 = MB^2$ equals the power of $M$ with respect to $\omega$.
The diagram
Drag any point and the whole construction follows; ▶ runs a point along its circle and back. Tap a point to pin it, double-click to go back to the start. Effects redraws the figure in grayscale for print, in colour, or with the givens in focus.