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]

\(\mathbf{2a}\) \((031)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&0.170220055912&{}:{}&0.345741643370&{}:{}&0.484038300718&,\\B^\prime&{}\approx{}&-2.998426136278&{}:{}&-2.997901515037&{}:{}&6.996327651315&,\\C^\prime&{}\approx{}&-0.011115016909&{}:{}&0.018525028182&{}:{}&0.992589988727&. \end{alignedat} \]
2a (031)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.622334958066\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}8.995278408834\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.033345050727\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
2a (031)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.027772123623&{}:{}&0.069116118911&{}:{}&0.958656004711&. \end{alignedat} \]
2a (031)

Hiroyasu Kamo