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]

\(\mathbf{6a}\) \((231)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&1.029779944088&{}:{}&-0.012408310037&{}:{}&-0.017371634051&,\\B^\prime&{}\approx{}&5.998426136278&{}:{}&8.997901515037&{}:{}&-13.996327651315&,\\C^\prime&{}\approx{}&0.611115016909&{}:{}&-1.018525028182&{}:{}&1.407410011273&. \end{alignedat} \]
6a (231)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}-0.022334958066\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-17.995278408834\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.833345050727\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
6a (231)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&1.072016560229&{}:{}&-0.047861767358&{}:{}&-0.024154792871&. \end{alignedat} \]
6a (231)

Hiroyasu Kamo