Cut-The-Knot Action 18!

Creation of this applet was inspired by a tweet from Alexander Bogomolny. Shown below is a rectangular hyperbola. A rectangular hyperbola is one whose semimajor and semiminor axes are equal. Thus, its eccentricity . Suppose a triangle has 3 vertices that lie on such a hyperbola. How can we prove its orthocenter always lies on this hyperbola as well?

Quick (Silent) Demo