Derousseau's Generalization of the Malfatti circles

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


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

\(\mathbf{7}\) \((222)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.661796968651\overrightarrow{AI},\\\overrightarrow{BB^\prime}&\approx{}4.403485812745\overrightarrow{BI},\\\overrightarrow{CC^\prime}&\approx{}6.993790794476\overrightarrow{CI}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.448502526124&{}:{}&0.264718787460&{}:{}&0.286778686416&,\\B^\prime&\approx{}&0.733914302124&{}:{}&-1.642091487647&{}:{}&1.908177185523&,\\C^\prime&\approx{}&1.165631799079&{}:{}&2.797516317790&{}:{}&-2.963148116870&.\end{alignedat}\]
7 (222)

Hiroyasu Kamo