Cyclic Quadrangle

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 $\Delta A_2A_3A_4$, 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 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.

The Simson Line of $A_4$ with regard to Triangle $\Delta A_1A_2A_3$ is the same as that of $H_4$ with regard to the Triangle $\Delta H_1A_2A_3$.

See also Angle Bisector, Concyclic, Cyclic Polygon, Cyclic Quadrilateral, Orthocentric System

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 251-253, 1929.