Incenter-Excenter Circle

Given a triangle $\Delta A_1A_2A_3$, the points $A_1$, $I$, and $J_1$ lie on a line, where $I$ is the Incenter and $J_1$ is the Excenter corresponding to $A_1$. Furthermore, the Circle with $IJ_1$ as the Diameter has $P$ as its center, where $P$ is the intersection of $A_1J_1$ with the Circumcircle of $\Delta A_1A_2A_3$, and passes through $A_2$ and $A_3$. This Circle has Radius

$$r={\textstyle{1\over 2}}a_1\sec({\textstyle{1\over 2}}\alpha_1)=2R\sin({\textstyle{1\over 2}}\alpha_1).$$

It arises because $IJ_1J_2J_3$ forms an Orthocentric System.

See also Circumcircle, Excenter, Excenter-Excenter Circle, Incenter, Orthocentric System

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 185, 1929.