Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{3b}\) \((123)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-13}{4}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+13}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+13}{20}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-37}{24}&{}:{}&-\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-13}{2}&{}:{}&\frac{13\left(2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-13\right)}{24}&,\\B^\prime&={}&\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+13}{36}&{}:{}&-\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+3}{10}&{}:{}&\frac{13\left(2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+13\right)}{180}&,\\C^\prime&={}&-\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+13}{24}&{}:{}&\frac{2\sqrt{26}+5\sqrt{13}+10\sqrt{2}+13}{10}&{}:{}&-\frac{14\sqrt{26}+35\sqrt{13}+70\sqrt{2}-29}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-1.671895431601\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.299061765796\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-2.768396551412\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.278649238600&{}:{}&3.343790863202&{}:{}&-3.622440101802&,\\B^\prime&\approx{}&-0.249218138163&{}:{}&1.897185297387&{}:{}&-0.647967159224&,\\C^\prime&\approx{}&-2.306997126177&{}:{}&5.536793102824&{}:{}&-2.229795976647&.\end{alignedat}\]
3b (123)

Hiroyasu Kamo