Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{4}\) \((200)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{2\sqrt{26}+\sqrt{13}+2\sqrt{2}-17}{20}\overrightarrow{AI},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}+17}{6}\overrightarrow{BI},&\overrightarrow{CC^\prime}&={}\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}+17}{4}\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{2\sqrt{26}+\sqrt{13}+2\sqrt{2}+7}{24}&{}:{}&-\frac{2\sqrt{26}+\sqrt{13}+2\sqrt{2}-17}{50}&{}:{}&-\frac{13\left(2\sqrt{26}+\sqrt{13}+2\sqrt{2}-17\right)}{600}&,\\B^\prime&={}&\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}+17}{36}&{}:{}&-\frac{2\sqrt{26}+\sqrt{13}-2\sqrt{2}+7}{10}&{}:{}&\frac{13\left(2\sqrt{26}+\sqrt{13}-2\sqrt{2}+17\right)}{180}&,\\C^\prime&={}&\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}+17}{24}&{}:{}&\frac{2\sqrt{26}-\sqrt{13}+2\sqrt{2}+17}{10}&{}:{}&-\frac{34\sqrt{26}-17\sqrt{13}+34\sqrt{2}+169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.018399128630\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}4.662527196317\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}6.605228719117\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.984667392808&{}:{}&0.007359651452&{}:{}&0.007972955740&,\\B^\prime&\approx{}&0.777087866053&{}:{}&-1.797516317790&{}:{}&2.020428451737&,\\C^\prime&\approx{}&1.100871453186&{}:{}&2.642091487647&{}:{}&-2.742962940833&.\end{alignedat}\]
4 (200)

Hiroyasu Kamo