Creation of this applet was inspired by a [url=https://twitter.com/CutTheKnotMath/status/962335990303076352]tweet[/url] from [url=https://twitter.com/CutTheKnotMath]Alexander Bogomolny[/url]. [br][br]Shown below is a [b][color=#9900ff]rectangular hyperbola[/color][/b]. [br][b][color=#9900ff]A rectangular hyperbola is one whose semimajor and semiminor axes are equal. [/color][/b][br]Thus, its eccentricity [math]e=\sqrt{2}[/math]. [br][br]Suppose a [b][color=#bf9000]triangle[/color][/b] has 3 vertices that lie on such a [b][color=#9900ff]hyperbola[/color][/b]. [br]How can we prove its [b][color=#38761d]orthocenter[/color][/b] always lies on this [b][color=#9900ff]hyperbola[/color][/b] as well?