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