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Circle centered at orthocenter meets side of triangle

Source: 2022 Junior Macedonian Mathematical Olympiad P3

June 7, 2022

geometrygeometric transformationreflection

Problem Statement

Let

\triangle ABC

be an acute triangle with orthocenter

H

. The circle

\Gamma

with center

H

and radius

AH

meets the lines

AB

and

AC

at the points

E

and

F

respectively. Let

E'

F'

and

H'

be the reflections of the points

E

F

and

H

with respect to the line

BC

, respectively. Prove that the points

A

E'

F'

and

H'

lie on a circle.
Proposed by Jasna Ilieva