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tangent circumcircles wanted, symmedian related

Source: Ukrainian Geometry Olympiad 2017 XI p4

December 12, 2018

geometrysymmedianangle bisectortangent circlescircumcircle

Problem Statement

Let

AD

be the inner angle bisector of the triangle

ABC

. The perpendicular on the side

BC

at the point

D

intersects the outer bisector of

\angle CAB

at point

I

. The circle with center

I

and radius

ID

intersects the sides

AB

and

AC

at points

\Delta BXC

and

E

respectively.

A

-symmedian of

\Delta AFE

intersects the circumcircle of

\Delta AFE

again at point

X

. Prove that the circumcircles of

\Delta AFE

and

\Delta BXC

are tangent.

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