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Concurrent circles

Source: Mediterranean Mathematics Competition 2005, Problem 2

January 4, 2006

geometrycircumcirclegeometry proposed

Problem Statement

Let

k

and

k'

be concentric circles with center

O

and radius

R

and

R'

where

R<R'

holds. A line passing through

O

intersects

k

at

A

and

k'

at

B

where

O

is between

A

and

B

. Another line passing through

O

and distict from

AB

intersects

k

at

E

and

k'

at

F

where

E

is between

O

and

F

. Prove that the circumcircles of the triangles

OAE

and

OBF

, the circle with diameter

EF

and the circle with diameter

AB

are concurrent.

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