Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{1b}\) \((103)\)

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,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}0.171895431601\overrightarrow{AI_B},\\\overrightarrow{BB^\prime}&\approx{}0.499061765796\overrightarrow{BI_B},\\\overrightarrow{CC^\prime}&\approx{}2.468396551412\overrightarrow{CI_B}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&0.971350761400&{}:{}&-0.343790863202&{}:{}&0.372440101802&,\\B^\prime&\approx{}&0.415884804830&{}:{}&-0.497185297387&{}:{}&1.081300492557&,\\C^\prime&\approx{}&2.056997126177&{}:{}&-4.936793102824&{}:{}&3.879795976647&.\end{alignedat}\]
1b (103)

Hiroyasu Kamo