Mathematics

The key insight is that a rectangular hyperbola has a special symmetry property: its asymptotes are perpendicular, which means the hyperbola is invariant under a $90°$ rotation about its center. When a triangle's vertices lie on this curve, the orthocenter—found by reflecting each vertex across the opposite side's perpendicular—inherits this same rotational symmetry. Rather than computing coordinates directly, you can use the fact that perpendicularity and the hyperbola's geometric constraints force the orthocenter to satisfy the same equation as the vertices, placing it back on the curve.

The key insight is that a rectangular hyperbola has a special symmetry property: its asymptotes are perpendicular, which means the hyperbola is invariant under a $90°$ rotation about its center. When a triangle's vertices lie on this curve, the orthocenter—found by reflecting each vertex across the opposite side's perpendicular—inherits this same rotational symmetry. Rather than computing coordinates directly, you can use the fact that perpendicularity and the hyperbola's geometric constraints force the orthocenter to satisfy the same equation as the vertices, placing it back on the curve.