Is there a geometric proof to the statement below?
If $P$ is a point on a rectangular hyperbola with center $C$ and foci $S$ and $S'$, then $SP \times S'P=(CP)^2$.
I am staring at the triangle $\Delta PS'S$ and the cevian $PC$. The expression above looks like it might have arrived from two similar triangles but alas $\Delta PS'C$ and $\Delta PSC$ don't look similar. I was also thinking about reflection property but don't know how it will be useful.
I know how to prove this analytically. I was wondering if there is a pure geometric way of proving it.