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rove that the quadrilateral $O_1O_2O_3O_4$ is a parallelogram.

Source: Moldova TST 1993

August 8, 2023

geometryparallelogramcircumcircle

Problem Statement

Inside the parallelogram

ABCD

points

M, N, K

and

L{}

are on sides

AB, BC, CD{}

and

DA

, respectively. Let

O_1, O_2, O_3

and

O_4

be the circumcenters of triangles repesctively

MBN, NCK, KDL

and

LAM{}

. Prove that the quadrilateral

O_1O_2O_3O_4

is a parallelogram.