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Incentre of ABC, orthocentre of DEF,midpointof Arc collinear

Source: European Girls’ Mathematical Olympiad - DAY 1 - P2

April 12, 2014

incentercircumcirclereflectionhomothetyInversionEGMOEGMO 2014

Problem Statement

Let

D

and

E

be points in the interiors of sides

AB

and

AC

, respectively, of a triangle

ABC

, such that

DB = BC = CE

. Let the lines

CD

and

BE

meet at

F

. Prove that the incentre

I

of triangle

ABC

, the orthocentre

H

of triangle

DEF

and the midpoint

M

of the arc

BAC

of the circumcircle of triangle

ABC

are collinear.