A problem related to locus in circles Two circles intersect at $A$ and $B$; $PQ$ is a straight line through $A$ meeting the circles at $P$ and $Q$. Find the locus of midpoint of $PQ$.
I know the locus is a circle. But I am unable to prove it.

Angles $\angle BPA$ and $\angle BQA$ don't depend on the positions of $P$ and $Q$, while they vary on the outer parts of the circles. That means that all triangles $BQP$ have the same angles, independent of the positions of $P$ and $Q$, and that is true even if $P$ and $Q$ lie on the inner parts of the circles. 
If $M$ is the midpoint of $PQ$, it follows that median $BM$ forms a constant angle $\alpha$ with $PQ$. We then have $\angle AMB=\alpha$ if $PM<PA$, and $\angle AMB=\pi-\alpha$ if $PM>PA$: point $M$ lies then on the circle having $AB$ as a chord, subtended by an angle $\alpha$.
In particular, when $PQ$ is perpendicular to $AB$, the centers of the given circles are the midpoints of $BP$ and $BQ$, while the center of the locus is the midpoint of $BM$. It follows that the center of the locus is the midpoint of the centers of the given circles.
