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]

\(\mathbf{6}\) \((220)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.671364425571&{}:{}&0.136931489345&{}:{}&0.191704085083&,\\B^\prime&{}\approx{}&0.302831959204&{}:{}&-0.009439864014&{}:{}&0.706607904810&,\\C^\prime&{}\approx{}&2.972460320447&{}:{}&4.954100534079&{}:{}&-6.926560854526&. \end{alignedat} \]
6 (220)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.410794468036\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}1.514159796021\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}14.862301602236\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
6 (220)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.635649016855&{}:{}&0.346533535314&{}:{}&0.017817447831&. \end{alignedat} \]
6 (220)

Hiroyasu Kamo