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Collinearity with altitudes and circumcircles in the Centro

Source: Centroamerican and Caribbean Math Olympiad 2024 P3

October 18, 2024

geometrycollinearitycircumcircle

Problem Statement

Let

ABC

be a triangle,

H

its orthocenter, and

\Gamma

its circumcircle. Let

J

be the point diametrically opposite to

A

on

\Gamma

. The points

D

,

E

and

F

are the feet of the altitudes from

A

,

B

and

C

, respectively. The line

AD

intersects

\Gamma

again at

P

. The circumcircle of

EFP

intersects

\Gamma

again at

Q

. Let

K

be the second point of intersection of

JH

with

\Gamma

. Prove that

K

,

D

and

Q

are collinear.

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