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APMO 2010 P4 but you do it on all 3 sides

Source: IMO Shortlist 2023 G7

July 17, 2024

IMO Shortlistgeometry

Problem Statement

Let

ABC

be an acute, scalene triangle with orthocentre

H

. Let

\ell_a

be the line through the reflection of

B

with respect to

\ell_a

and the reflection of

C

with respect to

BH

. Lines

\ell_b

and

\ell_c

are defined similarly. Suppose lines

\ell_a

,

\ell_b

, and

\ell_c

determine a triangle

\mathcal T

.
Prove that the orthocentre of

\mathcal T

, the circumcentre of

\mathcal T

, and

H

are collinear.
Fedir Yudin, Ukraine

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