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circumcenter of a triangle lies on a given circle

Source: Greece JBMO TST 2008 p1

April 29, 2019

geometryCircumcentertangentisoscelescircle

Problem Statement

Given a point

A

that lies on circle

c(o,R)

(with center

O

and radius

R

). Let

(e)

be the tangent of the circle

B

at point

O

and a line

A

that passes through point

O

and intersects

(e)

at point

M

and the circle at points

B,C

(let

B

lie between

O

and

A

). If

AM = R\sqrt3

, prove that a) triangle

AMC

is isosceles. b) circumcenter of triangle

AMC

lies on circle

c

.

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