Derousseau's Generalization of the Malfatti circles

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


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

\(\mathbf{4b}\) \((301)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+7}{4}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7}{20}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}-17}{24}&{}:{}&-\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+7}{2}&{}:{}&\frac{13\left(2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+7\right)}{24}&,\\B^\prime&={}&\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7}{36}&{}:{}&-\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-17}{10}&{}:{}&\frac{13\left(2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7\right)}{180}&,\\C^\prime&={}&-\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7}{24}&{}:{}&\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7}{10}&{}:{}&-\frac{14\sqrt{26}-35\sqrt{13}+70\sqrt{2}-169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}12.341982757059\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.236121992692\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.034379086320\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&-1.056997126177&{}:{}&-24.683965514118&{}:{}&26.740962640295&,\\B^\prime&\approx{}&0.196768327244&{}:{}&0.291634021923&{}:{}&0.511597650834&,\\C^\prime&\approx{}&0.028649238600&{}:{}&-0.068758172640&{}:{}&1.040108934040&.\end{alignedat}\]
4b (301)

Hiroyasu Kamo