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]

\(\mathbf{3}\) \((022)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.321093475754&{}:{}&0.550455614898&{}:{}&0.770637860857&,\\B^\prime&{}\approx{}&0.016225903608&{}:{}&0.945913654641&{}:{}&0.037860441752&,\\C^\prime&{}\approx{}&0.739430042465&{}:{}&1.232383404108&{}:{}&-0.971813446573&. \end{alignedat} \]
3 (022)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.651366844693\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.081129518039\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.697150212324\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
3 (022)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.014242493332&{}:{}&0.854025234005&{}:{}&0.131732272663&. \end{alignedat} \]
3 (022)

Hiroyasu Kamo