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]

\(\mathbf{2}\) \((020)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.297275769760&{}:{}&0.540531570734&{}:{}&0.756744199027&,\\B^\prime&{}\approx{}&0.005507935911&{}:{}&0.981640213631&{}:{}&0.012851850458&,\\C^\prime&{}\approx{}&1.485840203979&{}:{}&2.476400339964&{}:{}&-2.962240543943&. \end{alignedat} \]
2 (020)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}1.621594712201\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.027539679553\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}7.429201019893\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
2 (020)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.015613713290&{}:{}&0.936247947250&{}:{}&0.048138339461&. \end{alignedat} \]
2 (020)

Hiroyasu Kamo