think in apollonian circles!

Note that the locus of all possible $P$ is just the apollonius circle preserving $\frac{OB}{AB}$. We thus use this definition exclusively. Let $X$ be the center of this apollonian circle. First, let's tie in the fact that $AOB$ is isoceles. Note that $\angle{XBO} = \angle{BAX} = \angle{OBA}$ (as $XB$ is tangent to the circumcircle of $OBA$), so $OB$ bisects $\angle{ABX}$. Thus $\frac{AB}{AO} = \frac{XP}{XO}$. Let's now use this ratio. Note that $\frac{PQ}{PO} = \frac{AB}{AO} = \frac{XP}{XO}$. However, $XP$ is also tangent to the circumcircle of $OPA$, so we also find that $\frac{AP}{PO} = \frac{XP}{XO}$. This thus indicates that $AP = PQ$, and so we are done!