Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{5a}\) \((213)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.203565106639\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-9.166725253634\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-3.599055681767\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.271420142185&{}:{}&-0.113091725910&{}:{}&-0.158328416275&,\\B^\prime&\approx{}&3.055575084545&{}:{}&5.074100112726&{}:{}&-7.129675197271&,\\C^\prime&\approx{}&1.199685227256&{}:{}&-1.999475378759&{}:{}&1.799790151504&.\end{alignedat}\]
5a (213)

Hiroyasu Kamo