Consider two circles $C,C'$ in euclidean plane which intersect in exactly two points $Q,R$ and consider the line $QR$ through these points.
The claim is that a point point $P$ lies on the line $QR$ if and only if for any two lines $\ell,\ell'$ through $P$ with $C\cap\ell=\lbrace A,B \rbrace$ and $C'\cap\ell'=\lbrace A',B' \rbrace$ the following formula line segments hold $$\vert PA\vert\cdot \vert PB\vert=\vert PA'\vert\cdot \vert PB'\vert.$$
Unfortunatley I have no idea how to prove this claim. Could someone be so kind and help me?
Best wishes