RMM 2009 Problem 3

Given four points $A_1, A_2, A_3, A_4$ in the plane, no three collinear, such that $$A_1A_2 \cdot A_3 A_4 = A_1 A_3 \cdot A_2 A_4 = A_1 A_4 \cdot A_2 A_3,$$ denote by $O_i$ the circumcenter of $\triangle A_j A_k A_l$ with $\{i,j,k,l\} = \{1,2,3,4\}.$ Assuming $\forall i A_i \neq O_i ,$ prove that the four lines $A_iO_i$ are concurrent or parallel.

Nikolai Ivanov Beluhov, Bulgaria