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.