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incircle excenter midpoints

Source: Middle European Mathematical Olympiad T-6

September 21, 2014

geometrygeometric transformationreflectionhomothetyperpendicular bisectorgeometry proposed

Problem Statement

Let the incircle

k

of the triangle

ABC

touch its side

BC

D

. Let the line

AD

intersect

k

L \neq D

and denote the excentre of

ABC

opposite to

A

K

. Let

M

and

N

be the midpoints of

BC

and

KM

respectively.
Prove that the points

B, C, N,

and

L

are concyclic.