Geometry Geometry

Given four points A_{1}, A_{2}, A_{3}, A_{4} in the plane, no three collinear, such that \ A_{1}A_{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_{i}O_{i} are concurrent or parallel.