Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{0a}\) \((011)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.199685227256\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.335024370989\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.610695319916\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.599580303007&{}:{}&0.666491792920&{}:{}&0.933088510088&,\\B^\prime&\approx{}&-0.111674790330&{}:{}&0.851100279560&{}:{}&0.260574510770&,\\C^\prime&\approx{}&-0.203565106639&{}:{}&0.339275177731&{}:{}&0.864289928907&.\end{alignedat}\]
0a (011)

Hiroyasu Kamo