Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{1b}\) \((103)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{4\sqrt{7}-3\sqrt{3}-5}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{4\sqrt{7}+3\sqrt{3}+5}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}\frac{8\sqrt{7}+9\sqrt{3}-5}{18}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{4\sqrt{7}-3\sqrt{3}-10}{5}&{}:{}&-\frac{4\sqrt{7}-3\sqrt{3}-5}{2}&{}:{}&\frac{7\left(4\sqrt{7}-3\sqrt{3}-5\right)}{10}&,\\B^\prime&={}&\frac{4\sqrt{7}+3\sqrt{3}+5}{50}&{}:{}&-\frac{4\sqrt{7}+3\sqrt{3}-10}{15}&{}:{}&\frac{7\left(4\sqrt{7}+3\sqrt{3}+5\right)}{150}&,\\C^\prime&={}&\frac{8\sqrt{7}+9\sqrt{3}-5}{30}&{}:{}&-\frac{8\sqrt{7}+9\sqrt{3}-5}{18}&{}:{}&\frac{8\sqrt{7}+9\sqrt{3}+40}{45}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.193426410776\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.692638588899\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}1.764137097591\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.922629435690&{}:{}&-0.193426410776&{}:{}&0.270796975086&,\\B^\prime&\approx{}&0.415583153339&{}:{}&-0.385277177798&{}:{}&0.969694024458&,\\C^\prime&\approx{}&1.058482258555&{}:{}&-1.764137097591&{}:{}&1.705654839036&.\end{alignedat}\]
1b (103)

Hiroyasu Kamo