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Excenter-Excenter Circle


ExcenterExcenterCircle

Given a triangle DeltaA_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=1/2a_1csc(1/2alpha_1)=2Rcos(1/2alpha_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

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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.

Referenced on Wolfram|Alpha

Excenter-Excenter Circle

Cite this as:

Weisstein, Eric W. "Excenter-Excenter Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Excenter-ExcenterCircle.html

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