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A lies on the radical axis of BQX and CPX

Source: APMO 2024 P1

July 29, 2024

geometryAPMO

Problem Statement

Let

ABC

be an acute triangle. Let

BCED

be a point on side

AB

and

E

be a point on side

AC

such that lines

BC

and

DE

are parallel. Let

X

be an interior point of

BCED

. Suppose rays

DX

and

EX

meet side

BC

at points

P

and

Q

, respectively, such that both

P

and

Q

lie between

B

and

C

. Suppose that the circumcircles of triangles

BQX

and

CPX

intersect at a point

Y \neq X

. Prove that the points

A, X

, and

Y

are collinear.

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