Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{6a}\) \((231)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{4\sqrt{21}-5\sqrt{3}-9}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{4\sqrt{21}+5\sqrt{3}+9}{2}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{8\sqrt{21}-5\sqrt{3}+27}{30}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{8\sqrt{21}-10\sqrt{3}+27}{45}&{}:{}&-\frac{4\sqrt{21}-5\sqrt{3}-9}{54}&{}:{}&-\frac{7\left(4\sqrt{21}-5\sqrt{3}-9\right)}{270}&,\\B^\prime&={}&\frac{4\sqrt{21}+5\sqrt{3}+9}{6}&{}:{}&\frac{8\sqrt{21}+10\sqrt{3}+27}{9}&{}:{}&-\frac{7\left(4\sqrt{21}+5\sqrt{3}+9\right)}{18}&,\\C^\prime&={}&\frac{8\sqrt{21}-5\sqrt{3}+27}{90}&{}:{}&-\frac{8\sqrt{21}-5\sqrt{3}+27}{54}&{}:{}&\frac{8\sqrt{21}-5\sqrt{3}+162}{135}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.022334958066\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-17.995278408834\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-1.833345050727\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.029779944088&{}:{}&-0.012408310037&{}:{}&-0.017371634051&,\\B^\prime&\approx{}&5.998426136278&{}:{}&8.997901515037&{}:{}&-13.996327651315&,\\C^\prime&\approx{}&0.611115016909&{}:{}&-1.018525028182&{}:{}&1.407410011273&.\end{alignedat}\]
6a (231)

Hiroyasu Kamo