Prove that if the incircle of triangle $ABC$ touches side $BC$ at $D$ and the $A$-excircle touches side $BC$ at $D'$, then the midpoint of $BC$ is the midpoint of $DD'$.
This is an interesting property that I discovered when doing a few problems but the solutions didn't prove it. After drawing several triangles and their in- and excircles, it seems to be true that the midpoint of the intouch and and extouch points is in fact the midpoint of the side of the triangle.
As another question, if anyone can prove that the antipode of the intouch $D$ which I have labeled as $U$ in the triangle below is collinear with the exsimilicenter $A$ and $D'$ that would help. Finally, is it also true that $A,U,D',$ and $I_a$ are collinear or just $A,U,$ and $D'$?
Note: $I_a$ is the center of the $A$-exircle.