Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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[Guy]
[Lob & Richmond]
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\(\mathbf{3}\) \((022)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{21}+2\sqrt{7}-\sqrt{3}+17}{18}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17}{10}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13}{2}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{4\sqrt{21}+4\sqrt{7}-2\sqrt{3}-11}{45}&{}:{}&\frac{2\sqrt{21}+2\sqrt{7}-\sqrt{3}+17}{54}&{}:{}&\frac{7\left(2\sqrt{21}+2\sqrt{7}-\sqrt{3}+17\right)}{270}&,\\B^\prime&={}&-\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17}{50}&{}:{}&\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-2}{15}&{}:{}&-\frac{7\left(2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17\right)}{150}&,\\C^\prime&={}&-\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13}{10}&{}:{}&-\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13}{6}&{}:{}&\frac{8\sqrt{21}-8\sqrt{7}+4\sqrt{3}-37}{15}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.651366844693\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.081129518039\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.697150212324\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.321093475754&{}:{}&0.550455614898&{}:{}&0.770637860857&,\\B^\prime&\approx{}&0.016225903608&{}:{}&0.945913654641&{}:{}&0.037860441752&,\\C^\prime&\approx{}&0.739430042465&{}:{}&1.232383404108&{}:{}&-0.971813446573&.\end{alignedat}\]
3 (022)

Hiroyasu Kamo