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]

\(\mathbf{6c}\) \((330)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.811435789111&{}:{}&2.028589472778&{}:{}&-2.840025261889&,\\B^\prime&{}\approx{}&0.330216137896&{}:{}&1.440288183862&{}:{}&-0.770504321758&,\\C^\prime&{}\approx{}&0.302779483315&{}:{}&0.504632472192&{}:{}&0.192588044493&. \end{alignedat} \]
6c (330)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.405717894556\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.110072045965\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.100926494438\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
6c (330)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.572129290421&{}:{}&1.149659403542&{}:{}&-0.721788693962&. \end{alignedat} \]
6c (330)

Hiroyasu Kamo