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**)

\(\mathbf{7b}\) \((323)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-10.389578833482\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.012895094052\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.587641845578\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&5.155831533393&{}:{}&10.389578833482&{}:{}&-14.545410366875&,\\B^\prime&\approx{}&-0.007737056431&{}:{}&1.025790188103&{}:{}&-0.018053131672&,\\C^\prime&\approx{}&-0.352585107347&{}:{}&0.587641845578&{}:{}&0.764943261769&.\end{alignedat}\]
7b (323)

Hiroyasu Kamo