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# External Similitude Center

In general, the external similitude center of two circles and with centers given in Cartesian coordinates is given by

 (1)

In trilinear coordinates, the external center of similitude is given by , where

 (2) (3) (4)

The incircle and circumcircle of a triangle have two similitude centers, namely the internal similitude center Si and the external center of similitude Se. The external center of similitude of the circumcircle and incircle Se is the isogonal conjugate of the Nagel point of . It is Kimberling center and has equivalent triangle center functions

 (5) (6) (7)

The two points Si and Se share certain similar properties, but there seems to be no straightforward analogy between the two. For instance, the internal similitude center Si is the homothetic center of the tangential, intangents, and extangents triangles of triangle taken pairwise, but the only comparable property of Se is more complicated: Se is the homothetic center of the tangential triangle and the reflection of the intangents triangle in the incenter of .

The following table summarized the external similitude centers for a number of named circles.

Circle-Circle Tangents, Homothetic Center, Internal Similitude Center, Midcircle, Similitude Center

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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, 1929.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998.Kimberling, C. "Encyclopedia of Triangle Centers: X(56)=External Center of Similitude of Circumcircle and Incircle." http://faculty.evansville.edu/ck6/encyclopedia/ETC.html#X56.

## Referenced on Wolfram|Alpha

External Similitude Center

## Cite this as:

Weisstein, Eric W. "External Similitude Center." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ExternalSimilitudeCenter.html