Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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\(\mathbf{0c}\) \((110)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.198129682738\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.225398830309\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}3.697790979797\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.396259365476&{}:{}&0.990648413689&{}:{}&-1.386907779164&,\\B^\prime&\approx{}&0.676196490926&{}:{}&1.901595321235&{}:{}&-1.577791812161&,\\C^\prime&\approx{}&11.093372939391&{}:{}&18.488954898986&{}:{}&-28.582327838377&.\end{alignedat}\]
0c (110)

Hiroyasu Kamo