MathDB

Let

ABC

be a triangle with incenter

I

and let

\Gamma

be a circle centered at

I

, whose radius is greater than the inradius and does not pass through any vertex. Let

X_{1}

be the intersection point of

\Gamma

and line

AB

, closer to

B

;

X_{2}

X_{3}

the points of intersection of

\Gamma

and line

BC

, with

X_{2}

closer to

B

; and let

X_{4}

be the point of intersection of

\Gamma

with line

CA

closer to

C

. Let

K

be the intersection point of lines

X_{1}X_{2}

and

X_{3}X_{4}

. Prove that

AK

bisects segment

X_{2}X_{3}

2