Suppose we are given two points, say $A$ and $B$. Point $P$ is moving such that $$\frac{PA}{PB} = r \ne1,$$ where $r$ is constant. It is a well known result that the locus of $P$ is circle. This is one of the special cases of Circles of Apollonius.
I have a question in its proof. Please refer Proof Using Angle Bisector Theorem.
It is clear that every point on the locus lies on the circle with diameter $CD$ but the converse is not clear -- I couldn't show that every point on the circle with diameter $CD$ lies on the locus of $P$. I suppose similar triangles are useful here but I don't know how.
Please help. Thanks in advance!