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]

\(\mathbf{3c}\) \((132)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.394441033370&{}:{}&-1.513897416576&{}:{}&2.119456383207&,\\B^\prime&{}\approx{}&-18.488954898986&{}:{}&-23.651939865314&{}:{}&43.140894764300&,\\C^\prime&{}\approx{}&-0.405717894556&{}:{}&-0.676196490926&{}:{}&2.081914385482&. \end{alignedat} \]
3cā€‚(132)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.302779483315\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-6.162984966329\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.135239298185\overrightarrow{CI_C}. \end{aligned} \] \[ \begin{alignedat}{4} I_C&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&. \end{alignedat} \]
3cā€‚(132)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.623958723754&{}:{}&-1.838128524240&{}:{}&3.462087247994&. \end{alignedat} \]
3cā€‚(132)

Hiroyasu Kamo