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2 circles tangent to a third one wanted, one more circle given

Source: P3 Francophone Math Olympiad Junior 2023

May 2, 2023

geometrycircumcircletangent circles

Problem Statement

Let

\Gamma

and

\Gamma'

be two circles with centres

O

and

O'

, such that

O

belongs to

\Gamma'

. Let

M

be a point on

\Gamma'

, outside of

\Gamma

. The tangents to

\Gamma

that go through

M

touch

\Gamma

in two points

A

and

B

, and cross

\Gamma'

again in two points

C

and

D

. Finally, let

E

be the crossing point of the lines

AB

and

CD

. Prove that the circumcircles of the triangles

CEO'

and

DEO'

are tangent to

\Gamma'

.

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