Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{1c}\) \((112)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.913750933176\overrightarrow{AI_C},\\\overrightarrow{BB^\prime}&\approx{}-0.547195063607\overrightarrow{BI_C},\\\overrightarrow{CC^\prime}&\approx{}-2.251267278456\overrightarrow{CI_C}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.771562266706&{}:{}&-2.741252799527&{}:{}&2.969690532821&,\\B^\prime&\approx{}&-0.683993829509&{}:{}&-0.094390127214&{}:{}&1.778383956722&,\\C^\prime&\approx{}&-2.814084098071&{}:{}&-6.753801835369&{}:{}&10.567885933440&.\end{alignedat}\]
1c (112)

Hiroyasu Kamo