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A convex, non-cyclical quadrilateral, and two cyclical ones

Source: 38th Brazilian MO (2016) - Second Day, Problem 6

November 23, 2016

Brazilian Math Olympiad 2016geometrycircumcircleangle bisector

Problem Statement

Lei it

ABCD

be a non-cyclical, convex quadrilateral, with no parallel sides. The lines

AB

and

CD

meet in

E

. Let it

M \not= E

be the intersection of circumcircles of

ADE

and

BCE

. The internal angle bisectors of

ABCD

form an convex, cyclical quadrilateral with circumcenter

I

. The external angle bisectors of

ABCD

form an convex, cyclical quadrilateral with circumcenter

J

.
Show that

I,J,M

are colinear.