Length of Chord Projected from Point on Intersecting Circle

Theorem
Let $C_1$ and $C_2$ be two circles which intersect at $A$ and $B$.

Let $T$ be a point on $C_1$.

Let $P$ and $Q$ be the points $TA$ and $TB$ intersect $C_2$.


 * Chord-Projected-from-Point-on-Intersecting-Circle.png

Then $PQ$ is constant, wherever $T$ is positioned on $C_1$.

Proof
Outline of proof:

By angle in the same segment, $\angle APB$ and $\angle ATB$ are constant.

Hence $\angle PBQ$ is constant.

This forces $PQ$ to be constant.

However this must be divided into multiple cases as sometimes $P, Q, T$ are contained in $C_1$.

The limit cases $T = A$ and $T = B$ can also be proven by interpreting $AT, BT$ as tangents at $T$.