Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{7b}\) \((323)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+7}{4}\overrightarrow{AI_B},&\overrightarrow{BB^\prime}&={}\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7}{30}\overrightarrow{BI_B},&\overrightarrow{CC^\prime}&={}-\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7}{20}\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&-\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}-17}{24}&{}:{}&-\frac{2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+7}{2}&{}:{}&\frac{13\left(2\sqrt{26}-5\sqrt{13}-10\sqrt{2}+7\right)}{24}&,\\B^\prime&={}&\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7}{36}&{}:{}&-\frac{2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-17}{10}&{}:{}&\frac{13\left(2\sqrt{26}-5\sqrt{13}+10\sqrt{2}-7\right)}{180}&,\\C^\prime&={}&-\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7}{24}&{}:{}&\frac{2\sqrt{26}+5\sqrt{13}-10\sqrt{2}-7}{10}&{}:{}&-\frac{14\sqrt{26}+35\sqrt{13}-70\sqrt{2}-169}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-3.742963243466\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}-0.022919390880\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}-0.354182989039\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.623827207244&{}:{}&7.485926486933&{}:{}&-8.109753694177&,\\B^\prime&\approx{}&-0.019099492400&{}:{}&1.068758172640&{}:{}&-0.049658680240&,\\C^\prime&\approx{}&-0.295152490866&{}:{}&0.708365978077&{}:{}&0.586786512788&.\end{alignedat}\]
7b (323)

Hiroyasu Kamo