Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{4}\) \((200)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.015299821974\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.918870481961\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}11.302849787676\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.987760142421&{}:{}&0.005099940658&{}:{}&0.007139916921&,\\B^\prime&\approx{}&0.583774096392&{}:{}&-0.945913654641&{}:{}&1.362139558248&,\\C^\prime&\approx{}&2.260569957535&{}:{}&3.767616595892&{}:{}&-5.028186553427&.\end{alignedat}\]
4 (200)

Hiroyasu Kamo