Let be the reflections of over and define similarly, so it suffices to show the incenter of lies on .

WLOG the problem is in the configuration in my diagram, so that is on minor arc . It's not hard to prove by angle-chasing that if is on with , then is the midpoint of arc not containing , so instead of dealing with the incenter of we deal with the orthocenter of .

Let be the desired orthocenter and be the orthocenter of ; it's well-known that . Now by angle-chasing are inversely congruent, i.e. share a common perpendicular bisector , which a reflection about switches . Then clearly correspond in this reflection, so it suffices to show that , or alternatively that . But this is a well-known property of Simson lines (proof: angle chase), so we're done.

This solution was created by translating the previous complex bash into synthetic