Some notes on four-body co-circular central configurations

There are two symmetric families of four-body co-circular central configurations, namely the kite and isosceles trapezoid. Using mutual distances as coordinates, we prove that if the four-body central configuration is an isosceles trapezoid, then the diagonals of the isosceles trapezoid cannot be perpendicular to each other. Furthermore, we show that for any four-body co-circular central configuration, the diagonals of the quadrilateral cannot be perpendicular except that the configuration is a kite.