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An elegant geometry problem

Source: The Golden Digits Contest, April 2024, P3

April 21, 2024

geometry

Problem Statement

Let

ABC

be an acute scalene triangle with orthocentre

H{}

and circumcentre

O.{}

Let

P{}

be an arbitrary point on the segment

OH

and

O_a

be the circumcentre of

PBC.{}

The line

PO_a

intersects the line

HA

at

P{}

Define

X_b

and

X_c

similarly. Let

Q{}

be the isogonal conjugate of

P{}

and

X{}

be the circumcentre of

X_aX_bX_c.{}

Prove that

PQ

and

HX

are parallel.
Proposed by David Anghel

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