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Similar triangles ACD, AEF, AMN

Source: German Mathematical Competition BWM 2005, 2nd round, problem

September 1, 2005

geometrygeometric transformationrotationratiogeometry proposed

Problem Statement

Two circles

k_1

and

k_2

intersect at two points

A

and

B

. Some line through the point

B

meets the circle

k_1

at a point

C

(apart from

B

), and the circle

k_2

at a point

E

(apart from

B

). Another line through the point

B

meets the circle

k_1

at a point

D

(apart from

B

), and the circle

k_2

at a point

F

(apart from

B

). Assume that the point

B

lies between the points

C

and

E

and between the points

D

and

F

. Finally, let

M

and

N

be the midpoints of the segments

CE

and

DF

. Prove that the triangles

ACD

,

AEF

and

AMN

are similar to each other.

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