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# Orthocentric System

A set of four points, one of which is the orthocenter of the other three. In an orthocentric system, each point is the orthocenter of the triangle of the other three, as illustrated above (Coxeter and Greitzer 1967, p. 39). The incenter and excenters of a triangle are an orthocentric system.

The centers of the circumcircles of the points in an orthocentric system form another orthocentric system congruent to the first, and are the reflection of the original points in their common nine-point center (Wells 1991).

The centroids of the points in an orthocentric system form another orthocentric system similar to the first, but one third the size (Wells 1991).

The sum of the squares of any nonadjacent pair of connectors of an orthocentric system equals the square of the diameter of the circumcircle. Orthocentric systems are used to define orthocentric coordinates.

The four circumcircles of points in an orthocentric system taken three at a time (illustrated above) have equal radius (Wells 1991).

The four triangles of an orthocentric system have a common nine-point circle, illustrated above. Furthermore, this circle is tangent to the 16 incircles and excircles of the four triangles (Wells 1991).

Angle Bisector, Circumcircle, Cyclic Quadrangle, Johnson's Theorem, Johnson Triangle, Johnson-Yff Circles, Nine-Point Circle, Orthic Triangle, Orthocenter, Orthocentric Quadrangle, Orthocentric Quadrilateral, Polar Circle, Rectangular Hyperbola

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## References

Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 109-114, 1952.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., 1967.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165-176, 1929.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 165, 1991.

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Orthocentric System

## Cite this as:

Weisstein, Eric W. "Orthocentric System." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthocentricSystem.html