Let A_1, A_2, A_3, and A_4 be four points on a circle, and H_1, H_2, H_3, H_4 the orthocenters of triangles Δ A_2 A_3 A_4, etc. If, from the eight points, four with different subscripts are chosen such that three are from one set and the fourth from the other, these points form an orthocentric system. There are eight such systems, which are analogous to the six sets of orthocentric systems obtained using the feet of the angle bisectors, orthocenter, and polygon vertices of a generic triangle. On the other hand, if all the points are chosen from one set, or two from each set, with all different subscripts, the four points lie on a circle. There are four pairs of such circles, and eight points lie by fours on eight equal circles.

angle bisector | concyclic | cyclic polygon | cyclic quadrilateral | orthocentric system