Derousseau's Generalization of the Malfatti circles

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


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[Lob & Richmond]
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\(\mathbf{1}\) \((002)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.321045743823\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}2.205989895504\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.091995643151\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.100871453186&{}:{}&0.528418297529&{}:{}&0.572453155657&,\\B^\prime&\approx{}&0.367664982584&{}:{}&-0.323593937302&{}:{}&0.955928954718&,\\C^\prime&\approx{}&0.015332607192&{}:{}&0.036798257260&{}:{}&0.947869135548&.\end{alignedat}\]
1 (002)

Hiroyasu Kamo