Derousseau's Generalization of the Malfatti circles

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


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Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{26}-\sqrt{13}-2\sqrt{2}+13}{20}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}-\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}-13}{6}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}-13}{4}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{26}-\sqrt{13}-2\sqrt{2}-11}{24}&{}:{}&\frac{2\sqrt{26}-\sqrt{13}-2\sqrt{2}+13}{50}&{}:{}&\frac{13\left(2\sqrt{26}-\sqrt{13}-2\sqrt{2}+13\right)}{600}&,\\B^\prime&={}&-\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}-13}{36}&{}:{}&\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}-3}{10}&{}:{}&-\frac{13\left(2\sqrt{26}-\sqrt{13}+2\sqrt{2}-13\right)}{180}&,\\C^\prime&={}&-\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}-13}{24}&{}:{}&-\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}-13}{10}&{}:{}&\frac{34\sqrt{26}+17\sqrt{13}-34\sqrt{2}-101}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.838203031349\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.596514187255\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.506209205524\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.301497473876&{}:{}&0.335281212540&{}:{}&0.363221313584&,\\B^\prime&\approx{}&0.099419031209&{}:{}&0.642091487647&{}:{}&0.258489481144&,\\C^\prime&\approx{}&0.084368200921&{}:{}&0.202483682210&{}:{}&0.713148116870&.\end{alignedat}\]
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Hiroyasu Kamo