Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{2c}\) \((130)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.102779483315\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}6.051873855217\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.201905964852\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.205558966630&{}:{}&0.513897416576&{}:{}&-0.719456383207&,\\B^\prime&\approx{}&18.155621565652&{}:{}&25.207495420870&{}:{}&-42.363116986522&,\\C^\prime&\approx{}&0.605717894556&{}:{}&1.009529824259&{}:{}&-0.615247718815&.\end{alignedat}\]
2c (130)

Hiroyasu Kamo