Excenter-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 $J_2J_3$ as the diameter has $Q$ as its center, where $P$ is the intersection of $A_1J_1$ with the Circumcircle of $A_1A_2A_3$ and $Q$ is the point opposite $P$ on the Circumcircle. The circle with diameter $J_2J_3$ also passes through $A_2$ and $A_3$ and has radius

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

It arises because the points $I$, $J_1$, $J_2$, and $J_3$ form an Orthocentric System.

See also Excenter, Incenter-Excenter Circle, Orthocentric System

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 185-186, 1929.