Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{5}\) \((202)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.101241841105\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}4.938669571233\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}4.191015156744\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.915631799079&{}:{}&0.040496736442&{}:{}&0.043871464479&,\\B^\prime&\approx{}&0.823111595205&{}:{}&-1.963201742740&{}:{}&2.140090147534&,\\C^\prime&\approx{}&0.698502526124&{}:{}&1.676406062698&{}:{}&-1.374908588822&.\end{alignedat}\]
5 (202)

Hiroyasu Kamo