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]

\(\mathbf{7c}\) \((332)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.211435789111&{}:{}&-3.028589472778&{}:{}&4.240025261889&,\\B^\prime&{}\approx{}&-0.663549471230&{}:{}&0.115267371694&{}:{}&1.548282099536&,\\C^\prime&{}\approx{}&-0.102779483315&{}:{}&-0.171299138859&{}:{}&1.274078622174&. \end{alignedat} \]
7c (332)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.605717894556\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.221183157077\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.034259827772\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
7c (332)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.327264710139&{}:{}&-0.488590949150&{}:{}&1.815855659289&. \end{alignedat} \]
7c (332)

Hiroyasu Kamo