Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{0a}\) \((011)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{4}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13}{6}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}-11}{24}&{}:{}&\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13}{50}&{}:{}&\frac{13\left(3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13\right)}{600}&,\\B^\prime&={}&-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{16}&{}:{}&-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-23}{10}&{}:{}&\frac{13\left(3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13\right)}{80}&,\\C^\prime&={}&-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13}{24}&{}:{}&\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13}{10}&{}:{}&-\frac{21\sqrt{26}-7\sqrt{13}+21\sqrt{2}-211}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.204841683445\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.414992282281\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.489024658739\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.506052104307&{}:{}&0.722905010067&{}:{}&0.783147094240&,\\B^\prime&\approx{}&-0.103748070570&{}:{}&0.834003087088&{}:{}&0.269744983482&,\\C^\prime&\approx{}&-0.122256164685&{}:{}&0.293414795243&{}:{}&0.828841369441&.\end{alignedat}\]
0a (011)

Hiroyasu Kamo