Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{6c}\) \((330)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+29}{10}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}-\frac{6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-29}{18}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}-\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-31}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+34}{5}&{}:{}&\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+29}{2}&{}:{}&-\frac{7\left(6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+29\right)}{10}&,\\B^\prime&={}&-\frac{6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-29}{6}&{}:{}&-\frac{12\sqrt{21}-20\sqrt{7}+30\sqrt{3}-67}{9}&{}:{}&\frac{7\left(6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-29\right)}{18}&,\\C^\prime&={}&-\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-31}{10}&{}:{}&-\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-31}{6}&{}:{}&\frac{24\sqrt{21}+40\sqrt{7}-60\sqrt{3}-109}{15}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.405717894556\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.110072045965\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.100926494438\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.811435789111&{}:{}&2.028589472778&{}:{}&-2.840025261889&,\\B^\prime&\approx{}&0.330216137896&{}:{}&1.440288183862&{}:{}&-0.770504321758&,\\C^\prime&\approx{}&0.302779483315&{}:{}&0.504632472192&{}:{}&0.192588044493&.\end{alignedat}\]
6c (330)

Hiroyasu Kamo