Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{6}\) \((220)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.410794468036\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}1.514159796021\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}14.862301602236\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.671364425571&{}:{}&0.136931489345&{}:{}&0.191704085083&,\\B^\prime&\approx{}&0.302831959204&{}:{}&-0.009439864014&{}:{}&0.706607904810&,\\C^\prime&\approx{}&2.972460320447&{}:{}&4.954100534079&{}:{}&-6.926560854526&.\end{alignedat}\]
6 (220)

Hiroyasu Kamo