Let be a rectangular hyperbola with center and let be a chord of with midpoint Prove that: The asymptotes of are the angle bisectors of line and the line through parallel to

hi GrantStar.

Let be the points at infinity on and let meet the hyperbola at points From we get that both equal Thus the pencil is harmonic, but from part 1, lies on so the polar of passes through the pole of which is the point at infinity on Then is and are parallel to the asymptotes, and in particular are perpendicular to each other. Then the angle bisector harmonic configuration gives that bisect and and thus the asymptotes bisect and the line through parallel to and thus to