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]

\(\mathbf{6b}\) \((321)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&9.350893551327&{}:{}&20.877233878318&{}:{}&-29.228127429646&,\\B^\prime&{}\approx{}&-0.011551064408&{}:{}&1.038503548026&{}:{}&-0.026952483619&,\\C^\prime&{}\approx{}&-0.346228427385&{}:{}&0.577047378975&{}:{}&0.769181048410&. \end{alignedat} \]
6b (321)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-20.877233878318\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.019251774013\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.577047378975\overrightarrow{CI_B}. \end{aligned} \] \[ \begin{alignedat}{4} I_B&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&. \end{alignedat} \]
6b (321)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.044399942077&{}:{}&1.089426837486&{}:{}&-0.045026895409&. \end{alignedat} \]
6b (321)

Hiroyasu Kamo