It is easy to prove that triangle ABF is equal to triangle GEF.
So GE=AB=AC. Angle ABF= Angle GEF
EHD angle =90 degrees -HED angle. Angle BHD=90 degrees+angle HED
In quadrilateral ABHC, angle BHD+ angle ABH+ angle BAC+ angle HCA=360 degrees.
So angle HCA= 180 degrees-angle ABH- angle HED.
So angle ACD= angle ABH+ angle HED= angle GED.
So triangle GED is equal to triangle ACD.
Therefore, AD=GD and angle GDE= angle ADC.
So the ADG angle =90 degrees.
So the triangle ADG is an isosceles right triangle.
Because f is the midpoint of AG.
Therefore, it is easy to prove that FA=FD,FA⊥FD.
Hope to adopt