In triangle ABC let X be the point of tangency of the excircle opposite A with side BC. (A) Prove that the segment AX divides triangle ABC into two triangles, each having the same perimeter. (B) Prove or disprove: The point X must lie on the nine-point circle of triangle ABC.
I really need some help with parts (A) and (B). I think for (B) that it should be proved true because the three excircles and the incenter are all tangent to the nine-point circle by Feuerbach's theorem. Is this correct? How should I prove this using that theorem if that is the right direction to head?