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]

\(\mathbf{0}\) \((000)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.327040090657&{}:{}&0.280399962226&{}:{}&0.392559947117&,\\B^\prime&{}\approx{}&0.147886008493&{}:{}&0.507046638357&{}:{}&0.345067353150&,\\C^\prime&{}\approx{}&0.081129518039&{}:{}&0.135215863398&{}:{}&0.783654618563&. \end{alignedat} \]
0 (000)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.841199886679\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.739430042465\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.405647590195\overrightarrow{CI}. \end{aligned} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&. \end{alignedat} \]
0 (000)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&0.172965088465&{}:{}&0.293769570429&{}:{}&0.533265341105&. \end{alignedat} \]
0 (000)

Hiroyasu Kamo