MathDB

We are given an acute triangle

ABC

with

AB\neq AC

. Let

D

be a point of

BC

such that

DA

is tangent to the circumcircle of

ABC

. Let

E

and

F

be the circumcenters of triangles

ABD

and

ACD

, respectively, and let

M

be the midpoints

EF

. Prove that the line tangent to the circumcircle of

AMD

through

D

is also tangent to the circumcircle of

ABC

.
Proposed by Patrik Bak, Slovakia

Problem Statement