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]

\(\mathbf{4c}\) \((310)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&22.786745878783&{}:{}&54.466864696957&{}:{}&-76.253610575740&,\\B^\prime&{}\approx{}&0.171299138859&{}:{}&1.228398851812&{}:{}&-0.399697990670&,\\C^\prime&{}\approx{}&0.398129682738&{}:{}&0.663549471230&{}:{}&-0.061679153967&. \end{alignedat} \]
4c (310)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}10.893372939391\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.057099712953\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.132709894246\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
4c (310)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.477011451243&{}:{}&1.405234547637&{}:{}&-0.882245998880&. \end{alignedat} \]
4c (310)

Hiroyasu Kamo