Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{21}-2\sqrt{7}-\sqrt{3}+13}{18}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13}{10}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17}{2}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{4\sqrt{21}-4\sqrt{7}-2\sqrt{3}-19}{45}&{}:{}&\frac{2\sqrt{21}-2\sqrt{7}-\sqrt{3}+13}{54}&{}:{}&\frac{7\left(2\sqrt{21}-2\sqrt{7}-\sqrt{3}+13\right)}{270}&,\\B^\prime&={}&-\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13}{50}&{}:{}&\frac{2\sqrt{21}-2\sqrt{7}+\sqrt{3}+2}{15}&{}:{}&-\frac{7\left(2\sqrt{21}-2\sqrt{7}+\sqrt{3}-13\right)}{150}&,\\C^\prime&={}&-\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17}{10}&{}:{}&-\frac{2\sqrt{21}+2\sqrt{7}+\sqrt{3}-17}{6}&{}:{}&\frac{8\sqrt{21}+8\sqrt{7}+4\sqrt{3}-53}{15}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.841199886679\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.739430042465\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.405647590195\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.327040090657&{}:{}&0.280399962226&{}:{}&0.392559947117&,\\B^\prime&\approx{}&0.147886008493&{}:{}&0.507046638357&{}:{}&0.345067353150&,\\C^\prime&\approx{}&0.081129518039&{}:{}&0.135215863398&{}:{}&0.783654618563&.\end{alignedat}\]
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Hiroyasu Kamo