A' is the mid point of BC.
So BFEDC is a cyclic quadrilateral. Then .....?
Suppose triangle ABC is an acute angle triangle. Let the bases of the altitudes to B and C be E and F, respectively, and let A' be the midpoint of BC. Prove that triangle A'EF is isosceles.
Don't know how to start this one. Would the Euler Line come into play?