Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{6}\) \((220)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.178954256177\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.794010104496\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}7.408004356849\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.850871453186&{}:{}&0.071581702471&{}:{}&0.077546844343&,\\B^\prime&\approx{}&0.465668350749&{}:{}&-0.676406062698&{}:{}&1.210737711948&,\\C^\prime&\approx{}&1.234667392808&{}:{}&2.963201742740&{}:{}&-3.197869135548&.\end{alignedat}\]
6 (220)

Hiroyasu Kamo