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three "old" circles and four concurrent lines

Source: IMO Shortlist 2006, Geometry 6, AIMO 2007, TST 3, P3

June 28, 2007

ratiogeometryprojective geometryhomothetyIMO Shortlist

Problem Statement

Circles

w_{1}

and

w_{2}

with centres

O_{1}

and

O_{2}

are externally tangent at point

D

and internally tangent to a circle

w

at points

E

and

F

respectively. Line

t

is the common tangent of

w_{1}

and

w_{2}

at

D

. Let

AB

be the diameter of

w

perpendicular to

t

, so that

A, E, O_{1}

are on the same side of

t

. Prove that lines

AO_{1}

,

BO_{2}

,

EF

and

t

are concurrent.

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