Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{1c}\) \((112)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.398129682738\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.336509941420\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-3.631124313130\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.203740634524&{}:{}&-1.990648413689&{}:{}&2.786907779164&,\\B^\prime&\approx{}&-1.009529824259&{}:{}&-0.346039765679&{}:{}&2.355569589938&,\\C^\prime&\approx{}&-10.893372939391&{}:{}&-18.155621565652&{}:{}&30.048994505044&.\end{alignedat}\]
1c (112)

Hiroyasu Kamo