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Perpendicular Bisector of Angle Bisector, Concyclic Points

Source: Benelux Mathematical Olympiad 2011, Problem 2

May 7, 2011

geometryincentercircumcircleperpendicular bisectorcyclic quadrilateralangle bisector

Problem Statement

Let

ABC

be a triangle with incentre

I

. The angle bisectors

AI

BI

and

CI

meet

[BC]

[CA]

and

[AB]

D

E

and

F

, respectively. The perpendicular bisector of

[AD]

intersects the lines

BI

and

CI

M

and

N

, respectively. Show that

A

I

M

and

N

lie on a circle.