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collinear iff X is circumcenter Serbia Math Olyimpiad 2011

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April 8, 2011

geometrycircumcirclegeometry unsolved

Problem Statement

Let

H

be orthocenter and

O

circumcenter of an acuted angled triangle

ABC

.

D

and

E

are feets of perpendiculars from

A

and

B

on

BC

and

AC

respectively. Let

OD

and

OE

intersect

BE

and

AD

in

K

and

L

, respectively. Let

X

be intersection of circumcircles of

X

and

HLE

different than

H

, and

M

is midpoint of

AB

. Prove that

K, L, M

are collinear iff

X

is circumcenter of

EOD

.

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