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Simple triangle geometry [a fixed point]

Source: German TST 2004, IMO ShortList 2003, geometry problem 2

May 18, 2004

geometryIMO ShortlistFixed point

Problem Statement

Three distinct points

A

,

B

, and

C

are fixed on a line in this order. Let

\Gamma

be a circle passing through

A

and

C

whose center does not lie on the line

AC

. Denote by

P

the intersection of the tangents to

\Gamma

at

\Gamma

and

C

. Suppose

\Gamma

meets the segment

PB

at

Q

. Prove that the intersection of the bisector of

\angle AQC

and the line

AC

does not depend on the choice of

\Gamma

.

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