Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{5a}\) \((213)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.097804931748\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-5.112216644549\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-6.024208417227\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.122256164685&{}:{}&-0.058682959049&{}:{}&-0.063573205636&,\\B^\prime&\approx{}&1.278054161137&{}:{}&3.044886657820&{}:{}&-3.322940818957&,\\C^\prime&\approx{}&1.506052104307&{}:{}&-3.614525050336&{}:{}&3.108472946029&.\end{alignedat}\]
5a (213)

Hiroyasu Kamo