Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{5c}\) \((312)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+17}{6}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-17}{20}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-17}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}-7}{24}&{}:{}&-\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+17}{2}&{}:{}&\frac{13\left(3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+17\right)}{24}&,\\B^\prime&={}&\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-17}{16}&{}:{}&\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-7}{10}&{}:{}&-\frac{13\left(3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-17\right)}{80}&,\\C^\prime&={}&-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-17}{24}&{}:{}&-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-17}{10}&{}:{}&\frac{51\sqrt{26}-85\sqrt{13}+255\sqrt{2}-169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-11.923003058949\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.244419425875\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.049416853302\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-1.980750764737&{}:{}&-35.769009176847&{}:{}&38.749759941585&,\\B^\prime&\approx{}&-0.305524282344&{}:{}&0.511161148250&{}:{}&0.794363134093&,\\C^\prime&\approx{}&-0.061771066627&{}:{}&-0.148250559905&{}:{}&1.210021626533&.\end{alignedat}\]
5c (312)

Hiroyasu Kamo