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Miquel spam geo

Source: APMO 2024 P5

July 29, 2024

geometryAPMO 2024

Problem Statement

Line

\ell

intersects sides

BC

and

AD

of cyclic quadrilateral

ABCD

in its interior points

R

and

S

, respectively, and intersects ray

DC

beyond point

C

at

Q

, and ray

BA

beyond point

A

at

P

. Circumcircles of the triangles

QCR

and

QDS

intersect at

N \neq Q

, while circumcircles of the triangles

PAS

and

PBR

intersect at

M\neq P

. Let lines

AD

and

NQ

meet at point

X

, lines

KL

and

CD

meet at point

K

and lines

BC

and

AD

meet at point

L

. Prove that point

X

lies on line

KL

.

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