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IMO ShortList 2002, geometry problem 4

Source: IMO ShortList 2002, geometry problem 4

September 28, 2004

geometryIMO ShortlistcirclesCircumcenter

Problem Statement

Circles

S_1

and

S_2

intersect at points

P

and

Q

. Distinct points

A_1

and

B_1

(not at

P

or

Q

) are selected on

S_1

. The lines

A_1P

and

B_1P

meet

S_2

again at

A_2

and

B_2

respectively, and the lines

A_1B_1

and

A_2B_2

meet at

C

. Prove that, as

A_1

and

B_1

vary, the circumcentres of triangles

A_1A_2C

all lie on one fixed circle.

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