Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{4c}\) \((310)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13}{6}\overrightarrow{AI_C},&\overrightarrow{BB^\prime}&={}-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{20}\overrightarrow{BI_C},&\overrightarrow{CC^\prime}&={}\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13}{30}\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+37}{24}&{}:{}&\frac{3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13}{2}&{}:{}&-\frac{13\left(3\sqrt{26}+5\sqrt{13}+15\sqrt{2}+13\right)}{24}&,\\B^\prime&={}&-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13}{16}&{}:{}&-\frac{3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-23}{10}&{}:{}&\frac{13\left(3\sqrt{26}+5\sqrt{13}-15\sqrt{2}-13\right)}{80}&,\\C^\prime&={}&\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13}{24}&{}:{}&\frac{3\sqrt{26}-5\sqrt{13}+15\sqrt{2}-13}{10}&{}:{}&-\frac{51\sqrt{26}-85\sqrt{13}+255\sqrt{2}-341}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}11.256336392282\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}0.044419425875\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}0.182750186635\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&3.814084098071&{}:{}&33.769009176847&{}:{}&-36.583093274918&,\\B^\prime&\approx{}&0.055524282344&{}:{}&1.088838851750&{}:{}&-0.144363134093&,\\C^\prime&\approx{}&0.228437733294&{}:{}&0.548250559905&{}:{}&0.223311706801&.\end{alignedat}\]
4c (310)

Hiroyasu Kamo