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]
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\(\mathbf{6b}\) \((321)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{8\sqrt{7}+9\sqrt{3}+5}{2}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}-\frac{8\sqrt{7}-9\sqrt{3}-5}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{4\sqrt{7}-3\sqrt{3}+5}{18}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{8\sqrt{7}+9\sqrt{3}+10}{5}&{}:{}&\frac{8\sqrt{7}+9\sqrt{3}+5}{2}&{}:{}&-\frac{7\left(8\sqrt{7}+9\sqrt{3}+5\right)}{10}&,\\B^\prime&={}&-\frac{8\sqrt{7}-9\sqrt{3}-5}{50}&{}:{}&\frac{8\sqrt{7}-9\sqrt{3}+10}{15}&{}:{}&-\frac{7\left(8\sqrt{7}-9\sqrt{3}-5\right)}{150}&,\\C^\prime&={}&-\frac{4\sqrt{7}-3\sqrt{3}+5}{30}&{}:{}&\frac{4\sqrt{7}-3\sqrt{3}+5}{18}&{}:{}&-\frac{4\sqrt{7}-3\sqrt{3}-40}{45}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-20.877233878318\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.019251774013\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.577047378975\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&9.350893551327&{}:{}&20.877233878318&{}:{}&-29.228127429646&,\\B^\prime&\approx{}&-0.011551064408&{}:{}&1.038503548026&{}:{}&-0.026952483619&,\\C^\prime&\approx{}&-0.346228427385&{}:{}&0.577047378975&{}:{}&0.769181048410&.\end{alignedat}\]
6b (321)

Hiroyasu Kamo