Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{5b}\) \((303)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}2.242963243466\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.222919390880\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}0.054182989039\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.626172792756&{}:{}&-4.485926486933&{}:{}&4.859753694177&,\\B^\prime&\approx{}&0.185766159067&{}:{}&0.331241827360&{}:{}&0.482992013574&,\\C^\prime&\approx{}&0.045152490866&{}:{}&-0.108365978077&{}:{}&1.063213487212&.\end{alignedat}\]
5b (303)

Hiroyasu Kamo