Question 55
In triangle $ABC$, let $B'$ and $C'$ be the midpoints of sides $AC$ and $AB$, respectively, and let $H$ be the foot of the altitude from vertex $A$. Prove that the circumcircles of triangles $AB'C'$, $BC'H$ and $B'CH$ pass through a common point $I$, and that the line $HI$ bisects the segment $B'C'$.