Excenter

The center $J_i$ of an Excircle. There are three excenters for a given Triangle, denoted $J_1$, $J_2$, $J_3$. The Incenter $I$ and excenters $J_i$ of a Triangle are an Orthocentric System.

$$\overline{OI}\,{}^2+\overline{OJ_1}\,{}^2+\overline{OJ_2}\,{}^2+\overline{OJ_3}\,{}^2=12R^2,$$

where $O$ is the Circumcenter, $J_i$ are the excenters, and $R$ is the Circumradius (Johnson 1929, p. 190). Denote the Midpoints of the original Triangle $M_1$, $M_2$, and $M_3$. Then the lines $J_1M_1$, $J_2M_2$, and $J_3M_3$ intersect in a point known as the Mittenpunkt.

See also Centroid (Orthocentric System), Excenter-Excenter Circle, Excentral Triangle, Excircle, Incenter, Mittenpunkt

References

Dixon, R. Mathographics. New York: Dover, pp. 58-59, 1991.

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