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Prove that the four lines AiOi are concurrent or parallel

Source: Romanian Master in Mathematics 2009, Problem 3

March 7, 2009

geometrycircumcircle3D geometrytetrahedronprismrhombus

Problem Statement

Given four points

A_1, A_2, A_3, A_4

in the plane, no three collinear, such that A_1A_2 \cdot A_3 A_4 \equal{} A_1 A_3 \cdot A_2 A_4 \equal{} A_1 A_4 \cdot A_2 A_3, denote by

O_i

the circumcenter of

\triangle A_j A_k A_l

with \{i,j,k,l\} \equal{} \{1,2,3,4\}. Assuming

\forall i A_i \neq O_i ,

prove that the four lines

A_iO_i

are concurrent or parallel.
Nikolai Ivanov Beluhov, Bulgaria