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orthocenter wanted, circumcenters and BF = CE = BC given

Source: Balkan MO BMO Shortlist 2016 G3

July 5, 2019

geometrycircumcircleorthocenter

Problem Statement

Given that

ABC

is a triangle where

AB < AC

. On the half-lines

BA

and

CA

we take points

F

and

E

respectively such that

BF = CE = BC

. Let

M,N

and

H

be the mid-points of the segments

BF,CE

and

BC

respectively and

K

and

O

be the circumcenters of the triangles

ABC

and

MNH

respectively. We assume that

OK

cuts

BE

and

HN

at the points

A_1

and

B_1

respectively and that

C_1

is the point of intersection of

HN

and

FE

. If the parallel line from

A_1

to

OC_1

cuts the line

FE

at

D

and the perpendicular from

A_1

to the line

DB_1

cuts

FE

at the point

M_1

, prove that

E

is the orthocenter of the triangle

A_1OM_1

.

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