Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{2b}\) \((121)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{8\sqrt{7}-9\sqrt{3}+5}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}-\frac{8\sqrt{7}+9\sqrt{3}-5}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{4\sqrt{7}+3\sqrt{3}+5}{18}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{8\sqrt{7}-9\sqrt{3}+10}{5}&{}:{}&\frac{8\sqrt{7}-9\sqrt{3}+5}{2}&{}:{}&-\frac{7\left(8\sqrt{7}-9\sqrt{3}+5\right)}{10}&,\\B^\prime&={}&-\frac{8\sqrt{7}+9\sqrt{3}-5}{50}&{}:{}&\frac{8\sqrt{7}+9\sqrt{3}+10}{15}&{}:{}&-\frac{7\left(8\sqrt{7}+9\sqrt{3}-5\right)}{150}&,\\C^\prime&={}&-\frac{4\sqrt{7}+3\sqrt{3}+5}{30}&{}:{}&\frac{4\sqrt{7}+3\sqrt{3}+5}{18}&{}:{}&-\frac{4\sqrt{7}+3\sqrt{3}-40}{45}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-5.288776610198\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-1.058482258555\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-1.154397648165\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&3.115510644079&{}:{}&5.288776610198&{}:{}&-7.404287254278&,\\B^\prime&\approx{}&-0.635089355133&{}:{}&3.116964517109&{}:{}&-1.481875161976&,\\C^\prime&\approx{}&-0.692638588899&{}:{}&1.154397648165&{}:{}&0.538240940734&.\end{alignedat}\]
2b (121)

Hiroyasu Kamo