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]

\(\mathbf{5b}\) \((303)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-1.155831533393&{}:{}&-5.389578833482&{}:{}&7.545410366875&,\\B^\prime&{}\approx{}&0.207737056431&{}:{}&0.307543145230&{}:{}&0.484719798339&,\\C^\prime&{}\approx{}&0.019251774013&{}:{}&-0.032086290022&{}:{}&1.012834516009&. \end{alignedat} \]
5b (303)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}5.389578833482\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.346228427385\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.032086290022\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
5b (303)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.069300614852&{}:{}&-0.048163321605&{}:{}&0.978862706753&. \end{alignedat} \]
5b (303)

Hiroyasu Kamo