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O1, O2, O3 are collinear

Source: Indonesia IMO 2010 TST, Stage 1, Test 4, Problem 2

November 12, 2009

geometric transformationprojective geometrygeometrypower of a pointradical axisgeometry proposed

Problem Statement

Circles

\Gamma_1

and

\Gamma_2

are internally tangent to circle

\Gamma

at

P

and

Q

, respectively. Let

P_1Q_1

and

Q_1

are on

\Gamma_1

and

\Gamma_2

respectively such that

P_1Q_1

is the common tangent of

R

and

Q_1

. Assume that

\Gamma_1

and

\Gamma_2

intersect at

R

and

R_1

. Define

O_1,O_2,O_3

as the intersection of

PQ

and

P_1Q_1

, the intersection of

PR

and

P_1R_1

, and the intersection

QR

and

Q_1R_1

. Prove that the points

O_1,O_2,O_3

are collinear. Rudi Adha Prihandoko, Bandung

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