Orthocentroidal Circle


The orthocentroidal circle of a triangle DeltaABC is a central circle having the segment joining the triangle centroid G and orthocenter H of DeltaABC as its diameter (Kimberling 1998, p. 234). Since the Euler line passes through G and H, it therefore bisects the orthocentroidal circle.

It has circle function


which corresponds to the circumcenter O. The center of the circle is Kimberling center X_(381), which has equivalent triangle center functions


The circle has radius


where R is the circumradius of DeltaABC.

The circle does not pass through any notable centers other than G and H, which are Kimberling centers X_2 and X_4, respectively.

It is orthogonal to the Lester circle and Stevanović circle.

The orthocentroidal circle of any triangle always contains the incenter I (Guinand 1984). This is an interesting observation since it means that the incenter is always "close" to the Euler line of the triangle (although it does not lie on it).

See also

Euler Line

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Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl. Dublin: Hodges, Figgis, and Co., p. 215, 1888.Guinand, A. P. "Euler Lines, Tritangent Centers and Their Triangles." Amer. Math. Monthly 91, 290-300, 1984.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.

Referenced on Wolfram|Alpha

Orthocentroidal Circle

Cite this as:

Weisstein, Eric W. "Orthocentroidal Circle." From MathWorld--A Wolfram Web Resource.

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