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FM bisects the angle AFD?

Source: 2023 Girls in Mathematics Tournament- Level A- Problem 3

October 29, 2023

geometry

Problem Statement

Let

ABC

an acute triangle and

D

and

E

the feet of heights by

A

and

B

, respectively, and let

M

be the midpoint of

AC

. The circle that passes through

D

and

B

and is tangent to

BE

B

intersects the line

BM

F, F\neq B

. Show that

FM

is the angle bisector of

\angle AFD