Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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[Lob & Richmond]
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\(\mathbf{1}\) \((002)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.255872198631\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}1.485840203979\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.137698397764\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.004697758905&{}:{}&0.418624066210&{}:{}&0.586073692694&,\\B^\prime&\approx{}&0.297168040796&{}:{}&0.009439864014&{}:{}&0.693392095190&,\\C^\prime&\approx{}&0.027539679553&{}:{}&0.045899465921&{}:{}&0.926560854526&.\end{alignedat}\]
1 (002)

Hiroyasu Kamo