Prove that middle point of PQ is a circle

Question

Point $A$ is one of the points of intersection of two given intersecting circles. Any line is drawn through A to cut the circles again in P and Q. Prove that the locus of middle point of PQ is a circle.
My Attempt:
I was able to obtain the result with difficulty through analytic geometry but is there a geometrical solution to the problem

user10354138 · Accepted Answer

OK, let me give a quick analytic proof of my claim in the comments.
Choose polar coordinates with $A$ being the origin.  The general equation of circles through $A$ is $r=dcos(theta-theta_0)$ where $d$ is the diameter and $theta=theta_0$ is the half-ray containing the diameter through $A$.
So let $r=d_icos(theta-theta_i)$, $i=1,2$ be the two circles.  For any $lambdainmathbb{R}$, the locus of $P_lambda$ is therefore $$r=(1-lambda) d_1cos(theta-theta_1)+lambda d_2cos(theta-theta_2)$$
which yields the Cartesian version $(1-lambda)S_1+lambda S_2=0$.

Prove that middle point of PQ is a circle

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