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]

\(\mathbf{4b}\) \((301)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-5.350893551327&{}:{}&-15.877233878318&{}:{}&22.228127429646&,\\B^\prime&{}\approx{}&0.211551064408&{}:{}&0.294829785307&{}:{}&0.493619150285&,\\C^\prime&{}\approx{}&0.012895094052&{}:{}&-0.021491823420&{}:{}&1.008596729368&. \end{alignedat} \]
4b (301)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}15.877233878318\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.352585107347\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.021491823420\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
4b (301)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.048335533334&{}:{}&-0.049248256203&{}:{}&1.000912722870&. \end{alignedat} \]
4b (301)

Hiroyasu Kamo