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]

\(\mathbf{7b}\) \((323)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&5.155831533393&{}:{}&10.389578833482&{}:{}&-14.545410366875&,\\B^\prime&{}\approx{}&-0.007737056431&{}:{}&1.025790188103&{}:{}&-0.018053131672&,\\C^\prime&{}\approx{}&-0.352585107347&{}:{}&0.587641845578&{}:{}&0.764943261769&. \end{alignedat} \]
7b (323)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-10.389578833482\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.012895094052\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.587641845578\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
7b (323)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.029864174568&{}:{}&1.074264393877&{}:{}&-0.044400219308&. \end{alignedat} \]
7b (323)

Hiroyasu Kamo