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2 lines connecting incenters with excenters concurrent with angle bisector

Source: 2014 Oral Moscow Geometry Olympiad grades 10-11 p6

August 9, 2019

geometryincenterexcenterconcurrencyconcurrentangle bisector

Problem Statement

A convex quadrangle

ABCD

is given. Let

I

and

J

be the circles of circles inscribed in the triangles

ABC

and

ADC

, respectively, and

I_a

and

J_a

are the centers of the excircles circles of triangles

ABC

and

ADC

, respectively (inscribed in the angles

BAC

and

DAC

, respectively). Prove that the intersection point

K

of the lines

IJ_a

and

JI_a

lies on the bisector of the angle

BCD