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]

\(\mathbf{1c}\) \((112)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.203740634524&{}:{}&-1.990648413689&{}:{}&2.786907779164&,\\B^\prime&{}\approx{}&-1.009529824259&{}:{}&-0.346039765679&{}:{}&2.355569589938&,\\C^\prime&{}\approx{}&-10.893372939391&{}:{}&-18.155621565652&{}:{}&30.048994505044&. \end{alignedat} \]
1cā€‚(112)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.398129682738\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.336509941420\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-3.631124313130\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
1cā€‚(112)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-1.289807158786&{}:{}&-2.591789921755&{}:{}&4.881597080540&. \end{alignedat} \]
1cā€‚(112)

Hiroyasu Kamo