Also: Find all other points in the plane that are their own isogonal conjugates.
It seems quite obvious that in order for an isogonal conjugate to be itself it must also be the incenter, because it is constructed using angle bisectors - so angle bisectors would reflect to be themselves, but how would I show this formally?
Also, are there any other points on the plane that are their own isogonal conjugates? I would imagine only origins of excircles but I'm not sure how to prove it.