Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


[Other solutions]
[Guy]
[Lob & Richmond]
(0**)
(1**)
(2**)
(3**)

\(\mathbf{5c}\) \((312)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-11.093372939391\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.168210824064\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.066043227579\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-21.186745878783&{}:{}&-55.466864696957&{}:{}&77.653610575740&,\\B^\prime&\approx{}&-0.504632472192&{}:{}&0.327156703744&{}:{}&1.177475768448&,\\C^\prime&\approx{}&-0.198129682738&{}:{}&-0.330216137896&{}:{}&1.528345820634&.\end{alignedat}\]
5c (312)

Hiroyasu Kamo