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Tangency, intersections and linearity

Source: 2021 Turkey JBMO TST P8

May 24, 2021

geometrygeometry proposed

Problem Statement

w_1

and

w_2

circles have different diameters and externally tangent to each other at

X

. Points

A

and

B

are on

w_1

, points

C

and

D

are on

w_2

such that

AC

and

BD

are common tangent lines of these two circles.

CX

intersects

AB

at

AX \cap CD =H

and

w_1

at

F

second time.

(EFB)

intersects

AF

at

G

second time. If

AX \cap CD =H

, show that points

E, G, H

are collinear.

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