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Triangle similarity

Source: Rioplatense Olympiad 2011, Level 3, Problem 2

August 29, 2014

geometrycircumcirclepower of a pointradical axisgeometric transformationangle bisectorgeometry unsolved

Problem Statement

Let

ABC

an acute triangle and

H

its orthocenter. Let

E

and

F

be the intersection of lines

BH

and

CH

with

AC

and

AB

respectively, and let

D

be the intersection of lines

EF

and

BC

. Let

\Gamma_1

be the circumcircle of

AEF

, and

\Gamma_2

the circumcircle of

BHC

. The line

AD

intersects

\Gamma_1

at point

I \neq A

. Let

J

be the feet of the internal bisector of

\angle{BHC}

and

M

the midpoint of the arc

\stackrel{\frown}{BC}

from

\Gamma_2

that contains the point

H

. The line

MJ

intersects

\Gamma_2

at point

N \neq M

. Show that the triangles

EIF

and

CNB

are similar.

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