MathDB

Concentric Circles

Source: USAMO 1998

October 9, 2005

geometrycircumcircleperpendicular bisectorUSAMOpower of a point

Problem Statement

Let

{\cal C}_1

and

{\cal C}_2

be concentric circles, with

{\cal C}_2

in the interior of

{\cal C}_1

. From a point

A

on

{\cal C}_1

one draws the tangent

AB

to

{\cal C}_2

(

B\in {\cal C}_2

). Let

C

be the second point of intersection of

AB

and

{\cal C}_1

, and let

D

be the midpoint of

AB

. A line passing through

A

intersects

{\cal C}_2

at

E

and

F

in such a way that the perpendicular bisectors of

DE

and

CF

intersect at a point

M

on

AB

. Find, with proof, the ratio

AM/MC

.

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