Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{7a}\) \((233)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}-7}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7}{4}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7}{6}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+17}{24}&{}:{}&-\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}-7}{50}&{}:{}&-\frac{13\left(3\sqrt{26}-\sqrt{13}-3\sqrt{2}-7\right)}{600}&,\\B^\prime&={}&\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7}{16}&{}:{}&\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}+17}{10}&{}:{}&-\frac{13\left(3\sqrt{26}-\sqrt{13}+3\sqrt{2}+7\right)}{80}&,\\C^\prime&={}&\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7}{24}&{}:{}&-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+7}{10}&{}:{}&\frac{21\sqrt{26}+7\sqrt{13}-21\sqrt{2}+169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.014962219273\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-5.733536988108\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-3.609994854854\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.018702774091&{}:{}&-0.008977331564&{}:{}&-0.009725442528&,\\B^\prime&\approx{}&1.433384247027&{}:{}&3.293414795243&{}:{}&-3.726799042270&,\\C^\prime&\approx{}&0.902498713713&{}:{}&-2.165996912912&{}:{}&2.263498199199&.\end{alignedat}\]
7a (233)

Hiroyasu Kamo