Question 119
Let $ABC$ be a triangle. Let $A'$ be the center of the circle passing through the following three points: the midpoint of side $BC$, the projection of $B$ onto the internal bisector of $\angle ACB$, and the projection of $C$ onto the internal bisector of $\angle ABC$. The points $B'$ and $C'$ are defined analogously. Prove that the nine-point circle of triangle $ABC$ and the circumcircle of triangle $A'B'C'$ are concentric.