Excentral Triangle

The Triangle $J=\Delta J_1J_2J_3$ with Vertices corresponding to the Excenters of a given Triangle $A$, also called the Tritangent Triangle.

Beginning with an arbitrary Triangle $A$, find the excentral triangle $J$. Then find the excentral triangle $J'$ of that Triangle, and so on. Then the resulting Triangle $J^{(\infty)}$ approaches an Equilateral Triangle.

Call $T$ the Triangle tangent externally to the Excircles of $A$. Then the Incenter $I_T$ of $K$ coincides with the Circumcenter $C_J$ of Triangle $\Delta J_1J_2J_3$, where $J_i$ are the Excenters of $A$. The Inradius $r_T$ of the Incircle of $T$ is

$$r_T=2R+r={\textstyle{1\over 2}}(r+r_1+r_2+r_3),$$

where $R$ is the Circumradius of $A$, $r$ is the Inradius, and $r_i$ are the Exradii (Johnson 1929, p. 192).

See also Excenter, Excenter-Excenter Circle, Excircle, Mittenpunkt

References

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