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Straight line

Source: 41-st Vietnamese Mathematical Olympiad 2003

August 28, 2004

geometrygeometric transformationhomothetyinvarianttrigonometryblogsratio

Problem Statement

The circles

C_{1}

and

C_{2}

touch externally at

M

and the radius of

C_{2}

is larger than that of

C_{1}

.

A

is any point on

C_{2}

which does not lie on the line joining the centers of the circles.

B

and

C

are points on

C_{1}

such that

AB

and

AC

are tangent to

C_{1}

. The lines

BM

,

CM

intersect

C_{2}

again at

E

,

F

respectively.

D

is the intersection of the tangent at

A

and the line

EF

. Show that the locus of

D

as

A

varies is a straight line.

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