Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{3c}\) \((132)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.814731419583\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-3.576900917685\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-0.231463375738\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.796317145104&{}:{}&-2.444194258749&{}:{}&2.647877113645&,\\B^\prime&\approx{}&-4.471126147106&{}:{}&-6.153801835369&{}:{}&11.624927982475&,\\C^\prime&\approx{}&-0.289329219672&{}:{}&-0.694390127214&{}:{}&1.983719346886&.\end{alignedat}\]
3c (132)

Hiroyasu Kamo