Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{1a}\) \((013)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{8\sqrt{21}+5\sqrt{3}+27}{90}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}\frac{8\sqrt{21}-5\sqrt{3}-27}{6}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}\frac{4\sqrt{21}+5\sqrt{3}-9}{10}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{16\sqrt{21}+10\sqrt{3}-81}{135}&{}:{}&\frac{8\sqrt{21}+5\sqrt{3}+27}{162}&{}:{}&\frac{7\left(8\sqrt{21}+5\sqrt{3}+27\right)}{810}&,\\B^\prime&={}&-\frac{8\sqrt{21}-5\sqrt{3}-27}{18}&{}:{}&-\frac{16\sqrt{21}-10\sqrt{3}-81}{27}&{}:{}&\frac{7\left(8\sqrt{21}-5\sqrt{3}-27\right)}{54}&,\\C^\prime&={}&-\frac{4\sqrt{21}+5\sqrt{3}-9}{30}&{}:{}&\frac{4\sqrt{21}+5\sqrt{3}-9}{18}&{}:{}&-\frac{4\sqrt{21}+5\sqrt{3}-54}{45}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.803565106639\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.166725253634\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}1.799055681767\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.071420142185&{}:{}&0.446425059244&{}:{}&0.624995082941&,\\B^\prime&\approx{}&-0.055575084545&{}:{}&0.925899887274&{}:{}&0.129675197271&,\\C^\prime&\approx{}&-0.599685227256&{}:{}&0.999475378759&{}:{}&0.600209848496&.\end{alignedat}\]
1a (013)

Hiroyasu Kamo