Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{0b}\) \((101)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.270914945194\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}1.845597700941\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.448592648693\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.954847509134&{}:{}&-0.541829890387&{}:{}&0.586982381253&,\\B^\prime&\approx{}&1.537998084118&{}:{}&-4.536793102824&{}:{}&3.998795018706&,\\C^\prime&\approx{}&0.373827207244&{}:{}&-0.897185297387&{}:{}&1.523358090142&.\end{alignedat}\]
0b (101)

Hiroyasu Kamo