20 13 Solving the Finale of Junior Middle School Group (Grade 8) of Sichuan Mathematics League+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

Connect AF, extend to g, and make FG=AF. The purpose of expanding DC intersection is to connect eg and GD at H (not necessarily coincident with F).

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