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paralellogram wanted, 3 equal circles intersecing in pairs

Source: Switzerland - Swiss MO 2007 p6

July 18, 2020

geometryparallelogramequal circles

Problem Statement

Three equal circles

k_1, k_2, k_3

intersect non-tangentially at a point

P

. Let

A

and

B

be the centers of circles

k_1

and

k_2

. Let

D

and

C

be the intersection of

k_3

with

k_1

and

k_2

respectively, which is different from

P

. Show that

ABCD

is a parallelogram.