Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{2c}\) \((130)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{6}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13}{20}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}-\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-37}{24}&{}:{}&-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{2}&{}:{}&\frac{13\left(3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13\right)}{24}&,\\B^\prime&={}&\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13}{16}&{}:{}&\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+23}{10}&{}:{}&-\frac{13\left(3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13\right)}{80}&,\\C^\prime&={}&-\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13}{24}&{}:{}&-\frac{3\sqrt{26}-5\sqrt{13}-15\sqrt{2}+13}{10}&{}:{}&\frac{51\sqrt{26}-85\sqrt{13}-255\sqrt{2}+341}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.148064752916\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}3.376900917685\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.364796709071\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.037016188229&{}:{}&0.444194258749&{}:{}&-0.481210446978&,\\B^\prime&\approx{}&4.221126147106&{}:{}&7.753801835369&{}:{}&-10.974927982475&,\\C^\prime&\approx{}&0.455995886339&{}:{}&1.094390127214&{}:{}&-0.550386013553&.\end{alignedat}\]
2c (130)

Hiroyasu Kamo