Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{7c}\) \((332)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13}{6}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13}{20}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+37}{24}&{}:{}&\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13}{2}&{}:{}&-\frac{13\left(3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13\right)}{24}&,\\B^\prime&={}&-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13}{16}&{}:{}&-\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-23}{10}&{}:{}&\frac{13\left(3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13\right)XBw}{80}&,\\C^\prime&={}&\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{24}&{}:{}&\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{10}&{}:{}&-\frac{51\sqrt{26}+85\sqrt{13}-255\sqrt{2}-341}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-1.823983545356\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.274125279953\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.029612950583\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.544004113661&{}:{}&-5.471950636069&{}:{}&5.927946522408&,\\B^\prime&\approx{}&-0.342656599941&{}:{}&0.451749440095&{}:{}&0.890907159846&,\\C^\prime&\approx{}&-0.037016188229&{}:{}&-0.088838851750&{}:{}&1.125855039979&.\end{alignedat}\]
7c (332)

Hiroyasu Kamo