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]

\(\mathbf{1a}\) \((013)\)

Triangle connecting the centers of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} A^\prime&{}\approx{}&-0.071420142185&{}:{}&0.446425059244&{}:{}&0.624995082941&,\\B^\prime&{}\approx{}&-0.055575084545&{}:{}&0.925899887274&{}:{}&0.129675197271&,\\C^\prime&{}\approx{}&-0.599685227256&{}:{}&0.999475378759&{}:{}&0.600209848496&. \end{alignedat} \]
1a (013)

Angle bisectors

Approximately,
\[ \begin{aligned} \overrightarrow{AA^\prime}&\approx{}0.803565106639\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.166725253634\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}1.799055681767\overrightarrow{CI_A}. \end{aligned} \] \[ \begin{alignedat}{4} I_A&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&. \end{alignedat} \]
1a (013)

Radical circle of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime&{}\approx{}&-0.156518182688&{}:{}&0.768614941193&{}:{}&0.387903241495&. \end{alignedat} \]
1a (013)

Hiroyasu Kamo