Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{7}\) \((222)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.339626576009&{}:{}&0.275155593329&{}:{}&0.385217830661&,\\B^\prime&{}\approx{}&0.452113991507&{}:{}&-0.507046638357&{}:{}&1.054932646850&,\\C^\prime&{}\approx{}&2.918870481961&{}:{}&4.864784136602&{}:{}&-6.783654618563&. \end{alignedat} \]
7 (222)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.825466779988\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.260569957535\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}14.594352409805\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
7 (222)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.613777132768&{}:{}&0.334609751724&{}:{}&0.051613115508&. \end{alignedat} \]
7 (222)

Hiroyasu Kamo