Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{3a}\) \((033)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13}{4}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{6}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}-11}{24}&{}:{}&\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13}{50}&{}:{}&\frac{13\left(3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13\right)}{600}&,\\B^\prime&={}&-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13}{16}&{}:{}&-\frac{3\sqrt{26}-\sqrt{13}+3\sqrt{2}-23}{10}&{}:{}&\frac{13\left(3\sqrt{26}-\sqrt{13}+3\sqrt{2}-13\right)}{80}&,\\C^\prime&={}&-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{24}&{}:{}&\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{10}&{}:{}&-\frac{21\sqrt{26}+7\sqrt{13}-21\sqrt{2}-211}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.681628885940\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}0.733536988108\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}0.276661521521\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.147963892575&{}:{}&0.408977331564&{}:{}&0.443058775861&,\\B^\prime&\approx{}&-0.183384247027&{}:{}&0.706585204757&{}:{}&0.476799042270&,\\C^\prime&\approx{}&-0.069165380380&{}:{}&0.165996912912&{}:{}&0.903168467468&.\end{alignedat}\]
3a (033)

Hiroyasu Kamo