My grandmother, who used to be a math teacher, read this problem long ago and has always wondered about a proof. (She was never 100% sure that it was really true until I modelled the problem in Sketchpad for her. Now she is a believer.) I haven't tried to prove it yet, but told her I would throw it out to the wolves and see what people came up with.
Prove that the angles formed by connecting the feet of the altitudes of a triangle are themselves bisected by the altitudes.