This problem was really challenging to me , I've spent much time on this but maybe it is easy to you guys . Enjoy !
Triangle has the following property : there is an interior point P such that , , and . Prove that triangle is isosceles .
Moderator edit: Approved Challenge question.
Not sure I'll have any luck with that pure geometric solution though.
By the way there was a small typo in your post, in red above.