Derousseau's Generalization of the Malfatti circles

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


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[Guy]
[Lob & Richmond]
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\(\mathbf{2}\) \((020)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.398758158895\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.061330428767\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}3.308984843256\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.165631799079&{}:{}&0.559503263558&{}:{}&0.606128535521&,\\B^\prime&\approx{}&0.010221738128&{}:{}&0.963201742740&{}:{}&0.026576519133&,\\C^\prime&\approx{}&0.551497473876&{}:{}&1.323593937302&{}:{}&-0.875091411178&.\end{alignedat}\]
2 (020)

Hiroyasu Kamo