AoPS - Problem

Two points

$A$ and

$B$ lie on two branches of hyperbola given by equation

$y=\frac1x$ . Let

$A_x$ and

$A_y$ be projections of

$A$ onto coordinate axis, similarly, let

$B_x$ and

$B_y$ be projections of

$B$ onto coordinate axis. Prove that triangles

$AB_xB_y$ and

$BA_xA_y$ have equal areas.