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Incenters in cyclic quadrilaterals

Source: Romanian 2018 TST Problem 1 day 3

May 25, 2020

cyclic quadrilateralincentergeometry

Problem Statement

Let

ABCD

be a cyclic quadrilateral and let its diagonals

AC

and

BD

cross at

X

. Let

I

be the incenter of

XBC

, and let

J

be the center of the circle tangent to the side

BC

and the extensions of sides

AB

and

DC

beyond

B

and

C

. Prove that the line

IJ

bisects the arc

BC

of circle

ABCD

, not containing the vertices

A

and

D

of the quadrilateral.

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