Derousseau's Generalization of the Malfatti circles

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


[Other solutions]
[Guy]
[Lob & Richmond]
(0**)
(1**)
(2**)
(3**)

\(\mathbf{3}\) \((022)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}1.481600871370\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}0.337472803683\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}0.894771280883\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-0.234667392808&{}:{}&0.592640348548&{}:{}&0.642027044260&,\\B^\prime&\approx{}&0.056245467280&{}:{}&0.797516317790&{}:{}&0.146238214929&,\\C^\prime&\approx{}&0.149128546814&{}:{}&0.357908512353&{}:{}&0.492962940833&.\end{alignedat}\]
3 (022)

Hiroyasu Kamo