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Collinearity in a twist on cyclic quad + incenters diagram

Source: KöMaL A. 742

February 13, 2019

geometry

Problem Statement

Convex quadrilateral

ABCD

is inscribed in circle

\Omega

. Its sides

AD

and

BC

intersect at point

E

. Let

M

and

N

be the midpoints of the circle arcs

AB

and

CD

not containing the other vertices, and let

I

,

J

,

K

,

L

denote the incenters of triangles

ABD

,

ABC

,

BCD

,

CDA

, respectively. Suppose

\Omega

intersects circles

IJM

and

KLN

for the second time at points

U \neq M

and

V \neq N

. Show that the points

E

,

U

, and

V

are collinear.

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