Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{1a}\) \((013)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.764471598414\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.112216644549\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}2.690875083894\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.044410501982&{}:{}&0.458682959049&{}:{}&0.496906538969&,\\B^\prime&\approx{}&-0.028054161137&{}:{}&0.955113342180&{}:{}&0.072940818957&,\\C^\prime&\approx{}&-0.672718770973&{}:{}&1.614525050336&{}:{}&0.058193720637&.\end{alignedat}\]
1a (013)

Hiroyasu Kamo