Internal Similitude Center

In general, the internal similitude center of two circles C_1=C(x_1,r_1) and C_2=C(x_2,r_2) with centers given in Cartesian coordinates is given by

In trilinear coordinates, the internal center of similitude is given by alpha:beta:gamma, where

The incircle and circumcircle of a triangle DeltaABC have two similitude centers, namely the internal center of similitude Si and the external similitude center Se. The internal center of similitude of these two circles Si is the isogonal conjugate of the Gergonne point of DeltaABC. It is Kimberling center X_(55) and has equivalent triangle center functions

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

The following table summarizes the internal similitude centers for a number of named circles.

See also

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References

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Cite this as:

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

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