MathDB

Let

ABC

be an equiangular triangle with circumcircle

\omega

. Let point

F\in AB

and point

E\in AC

so that

\angle ABE+\angle ACF=60^{\circ}

. The circumcircle of triangle

AFE

intersects the circle

\omega

in the point

D

. The halflines

DE

and

DF

intersect the line through

B

and

C

in the points

X

and

Y

. Prove that the incenter of the triangle

DXY

is independent of the choice of

E

and

F

.
(The angles in the problem statement are not directed. It is assumed that

E

and

F

are chosen in such a way that the halflines

DE

and

DF

indeed intersect the line through

B

and

C

3