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**)

\(\mathbf{5b}\) \((303)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{4\sqrt{7}+3\sqrt{3}-5}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{4\sqrt{7}-3\sqrt{3}+5}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}\frac{8\sqrt{7}-9\sqrt{3}-5}{18}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{4\sqrt{7}+3\sqrt{3}-10}{5}&{}:{}&-\frac{4\sqrt{7}+3\sqrt{3}-5}{2}&{}:{}&\frac{7\left(4\sqrt{7}+3\sqrt{3}-5\right)}{10}&,\\B^\prime&={}&\frac{4\sqrt{7}-3\sqrt{3}+5}{50}&{}:{}&-\frac{4\sqrt{7}-3\sqrt{3}-10}{15}&{}:{}&\frac{7\left(4\sqrt{7}-3\sqrt{3}+5\right)}{150}&,\\C^\prime&={}&\frac{8\sqrt{7}-9\sqrt{3}-5}{30}&{}:{}&-\frac{8\sqrt{7}-9\sqrt{3}-5}{18}&{}:{}&\frac{8\sqrt{7}-9\sqrt{3}+40}{45}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}5.389578833482\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.346228427385\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.032086290022\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-1.155831533393&{}:{}&-5.389578833482&{}:{}&7.545410366875&,\\B^\prime&\approx{}&0.207737056431&{}:{}&0.307543145230&{}:{}&0.484719798339&,\\C^\prime&\approx{}&0.019251774013&{}:{}&-0.032086290022&{}:{}&1.012834516009&.\end{alignedat}\]
5b (303)

Hiroyasu Kamo