Two circles intersecting, common tangent

Question

Two circles intersecting in $A$ and $B$. They have a common tangent, with point $P$ on circle one and point $Q$ on circle two. Prove that the line through $A$ and $B$ cuts the line $PQ$ by half.

Answer 1

Let $R$ be the intersection of the line $PQ$ and the line $AB$. Then by the Tangent-Secant Theorem $$|PR|^2=|RA|\cdot|RB|=|RQ|^2.$$

Answer 2

Similarly in spirit to Robert Z's answer, invert about the intersection point with inversion circle passing through $P$. Then, the entire diagram looks the same under inversion, hence the distance $QR$ is the inversion radius, which is $PR$.