Derousseau's Generalization of the Malfatti circles

\(a:b:c=5:12:13\).


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\(\mathbf{4a}\) \((211)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}-7}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7}{4}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7}{6}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+17}{24}&{}:{}&-\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}-7}{50}&{}:{}&-\frac{13\left(3\sqrt{26}+\sqrt{13}+3\sqrt{2}-7\right)}{600}&,\\B^\prime&={}&\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7}{16}&{}:{}&\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+17}{10}&{}:{}&-\frac{13\left(3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7\right)}{80}&,\\C^\prime&={}&\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7}{24}&{}:{}&-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7}{10}&{}:{}&\frac{21\sqrt{26}-7\sqrt{13}+21\sqrt{2}+169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.538175016779\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-5.414992282281\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-3.822357992072\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.672718770973&{}:{}&-0.322905010067&{}:{}&-0.349813760906&,\\B^\prime&\approx{}&1.353748070570&{}:{}&3.165996912912&{}:{}&-3.519744983482&,\\C^\prime&\approx{}&0.955589498018&{}:{}&-2.293414795243&{}:{}&2.337825297225&.\end{alignedat}\]
4a (211)

Hiroyasu Kamo