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
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 $r=d\cos(\theta-\theta_0)$ where $d$ is the diameter and $\theta=\theta_0$ is the half-ray containing the diameter through $A$.
So let $r=d_i\cos(\theta-\theta_i)$, $i=1,2$ be the two circles. For any $\lambda\in\mathbb{R}$, the locus of $P_\lambda$ is therefore $$r=(1-\lambda) d_1\cos(\theta-\theta_1)+\lambda d_2\cos(\theta-\theta_2)$$ which yields the Cartesian version $(1-\lambda)S_1+\lambda S_2=0$.