Question 98
Let $ABC$ be a triangle. Let $A'$ be the centre of the circle through the midpoint of the side $BC$ and the orthogonal projections of $B$ and $C$ on the lines of support of the internal bisectrices of the angles $ACB$ and $ABC$, respectively; the points $B'$ and $C'$ are defined similarly. Prove that the nine-point circle of the triangle $ABC$ and the circumcircle of $A'B'C'$ are concentric.
RMM 2015 Shortlist, Mexico
RMM 2015 Shortlist, Mexico