Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{5}\) \((202)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.045071954466\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.972460320447\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}7.570798980107\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.963942436427&{}:{}&0.015023984822&{}:{}&0.021033578751&,\\B^\prime&\approx{}&0.594492064089&{}:{}&-0.981640213631&{}:{}&1.387148149542&,\\C^\prime&\approx{}&1.514159796021&{}:{}&2.523599660036&{}:{}&-3.037759456057&.\end{alignedat}\]
5 (202)

Hiroyasu Kamo