In geometry, an orthocentric system is a set of four points in the plane one of which is the orthocenter of the triangle formed by the other three.
If four points form an orthocentric system, then each of the four points is the orthocenter of the other three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles with the same circumradius.
Read more about Orthocentric System: The Common Nine-point Circle, The Common Orthic Triangle, Its Incenter and Excenters, The Orthocentric System and Its Orthic Axes, Euler Lines and Homothetic Orthocentric Systems, Further Properties
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... If an orthocentric system of four points A, B, C and H is given, then the four triangles formed by any combination of three distinct points of that system all share the ... and P be an arbitrary point in the plane of the orthocentric system ... The centers of the incircle and excircles of a triangle form an orthocentric system ...
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