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Coinciding orthocenters

Source: Bulgarian IMO TST 2004, Day 2, Problem 2

July 8, 2013

geometrycircumcirclegeometric transformationreflectionrotationtrapezoidpower of a point

Problem Statement

Let

H

be the orthocenter of

\triangle ABC

. The points

A_{1} \not= A

,

B_{1} \not= B

and

C_{1} \not= C

lie, respectively, on the circumcircles of

\triangle BCH

,

\triangle CAH

and

\triangle ABH

and satisfy

A_{1}H=B_{1}H=C_{1}H

. Denote by

H_{1}

,

H_{2}

and

H_{3}

the orthocenters of

\triangle A_{1}BC

,

\triangle B_{1}CA

and

\triangle C_{1}AB

, respectively. Prove that

\triangle A_{1}B_{1}C_{1}

and

\triangle H_{1}H_{2}H_{3}

have the same orthocenter.

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