Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{3b}\) \((123)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-5.193426410776\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.359305255565\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-2.319692653146\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&3.077370564310&{}:{}&5.193426410776&{}:{}&-7.270796975086&,\\B^\prime&\approx{}&-0.215583153339&{}:{}&1.718610511131&{}:{}&-0.503027357792&,\\C^\prime&\approx{}&-1.391815591888&{}:{}&2.319692653146&{}:{}&0.072122938741&.\end{alignedat}\]
3b (123)

Hiroyasu Kamo