Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{6b}\) \((321)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-13.841982757059\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.036121992692\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.334379086320\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&3.306997126177&{}:{}&27.683965514118&{}:{}&-29.990962640295&,\\B^\prime&\approx{}&-0.030101660577&{}:{}&1.108365978077&{}:{}&-0.078264317500&,\\C^\prime&\approx{}&-0.278649238600&{}:{}&0.668758172640&{}:{}&0.609891065960&.\end{alignedat}\]
6b (321)

Hiroyasu Kamo