Help proving a trianlge is isosceles

[chim richalds]

Hi. I'm seriously stuck on this problem. I don't even know where to begin.

Homework Statement

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.

Homework Equations

The Attempt at a Solution

I know that I have to show that A'E and A'F are the same length or that angles A'EF and A'FE are congruent in order to prove the triangle is isosceles. I'm clueless though.There's a typo in the topic title. Sorry.

 

[LCKurtz]

Hint: Draw the circle with diameter BC and observe the right angles are inscribed in a semicircle.

 

[zgozvrm]

First, show that A'F congruent to A'C. (Try drawing A'G such that G lies on CF and A'G is perpendicular to CF)

Use the same process to show that A'E congruent to A'C