Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{4c}\) \((310)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}10.893372939391\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.057099712953\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.132709894246\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&22.786745878783&{}:{}&54.466864696957&{}:{}&-76.253610575740&,\\B^\prime&\approx{}&0.171299138859&{}:{}&1.228398851812&{}:{}&-0.399697990670&,\\C^\prime&\approx{}&0.398129682738&{}:{}&0.663549471230&{}:{}&-0.061679153967&.\end{alignedat}\]
4c (310)

Hiroyasu Kamo