Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{7c}\) \((332)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+31}{10}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}\frac{6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-31}{18}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-29}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+26}{5}&{}:{}&-\frac{6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+31}{2}&{}:{}&\frac{7\left(6\sqrt{21}-10\sqrt{7}-15\sqrt{3}+31\right)}{10}&,\\B^\prime&={}&\frac{6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-31}{6}&{}:{}&\frac{12\sqrt{21}-20\sqrt{7}+30\sqrt{3}-53}{9}&{}:{}&-\frac{7\left(6\sqrt{21}-10\sqrt{7}+15\sqrt{3}-31\right)}{18}&,\\C^\prime&={}&\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-29}{10}&{}:{}&\frac{6\sqrt{21}+10\sqrt{7}-15\sqrt{3}-29}{6}&{}:{}&-\frac{24\sqrt{21}+40\sqrt{7}-60\sqrt{3}-131}{15}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.605717894556\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.221183157077\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.034259827772\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.211435789111&{}:{}&-3.028589472778&{}:{}&4.240025261889&,\\B^\prime&\approx{}&-0.663549471230&{}:{}&0.115267371694&{}:{}&1.548282099536&,\\C^\prime&\approx{}&-0.102779483315&{}:{}&-0.171299138859&{}:{}&1.274078622174&.\end{alignedat}\]
7c (332)

Hiroyasu Kamo