The centroid of the four points constituting an orthocentric system is the center of the common nine-point circle (Johnson 1929, p. 249). This fact automatically guarantees that the centroid of the incenter and excenters of a triangle is located at the circumcenter.
Orthocentric Centroid
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References
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, 1929.Referenced on Wolfram|Alpha
Orthocentric CentroidCite this as:
Weisstein, Eric W. "Orthocentric Centroid." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrthocentricCentroid.html