Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{2a}\) \((031)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.721998970971\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}4.036312625840\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.074811096366\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.097501286287&{}:{}&0.433199382582&{}:{}&0.469299331131&,\\B^\prime&\approx{}&-1.009078156460&{}:{}&-0.614525050336&{}:{}&2.623603206796&,\\C^\prime&\approx{}&-0.018702774091&{}:{}&0.044886657820&{}:{}&0.973816116272&.\end{alignedat}\]
2a (031)

Hiroyasu Kamo