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Help my diagram has too many points

Source: IMO Shortlist 2023 G6

July 17, 2024

IMO ShortlistgeometryISL 2023

Problem Statement

Let

ABC

be an acute-angled triangle with circumcircle

\omega

. A circle

\Gamma

is internally tangent to

\omega

at

A

and also tangent to

BC

at

D

. Let

AB

and

AC

intersect

\Gamma

at

P

and

Q

respectively. Let

M

and

N

be points on line

BC

such that

B

is the midpoint of

DM

and

C

is the midpoint of

DN

. Lines

MP

and

NQ

meet at

K

and intersect

\Gamma

again at

I

and

J

respectively. The ray

KA

meets the circumcircle of triangle

IJK

again at

X\neq K

.
Prove that

\angle BXP = \angle CXQ

.
Kian Moshiri, United Kingdom

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