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Circumradii of MEF and NEF are of same length

Source: France TST 2014 P2

June 14, 2014

geometrygeometric transformationreflectioncircumcirclerotationtrapezoidtrigonometry

Problem Statement

Two circles

O_1

and

O_2

intersect each other at

M

and

N

. The common tangent to two circles nearer to

M

touch

O_1

and

O_2

at

A

and

B

respectively. Let

O_2

and

D

be the reflection of

A

and

B

respectively with respect to

M

. The circumcircle of the triangle

DCM

intersect circles

O_1

and

O_2

respectively at points

E

and

F

(both distinct from

M

). Show that the circumcircles of triangles

MEF

and

NEF

have same radius length.

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