Derousseau's Generalization of the Malfatti circles

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


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\(\mathbf{6a}\) \((231)\)

Exactly,
\[\begin{aligned}\overrightarrow{AA^\prime}&={}-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{30}\overrightarrow{AI_A},&\overrightarrow{BB^\prime}&={}-\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13}{4}\overrightarrow{BI_A},&\overrightarrow{CC^\prime}&={}-\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13}{6}\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&={}&\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}+11}{24}&{}:{}&-\frac{3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13}{50}&{}:{}&-\frac{13\left(3\sqrt{26}+\sqrt{13}-3\sqrt{2}-13\right)}{600}&,\\B^\prime&={}&\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13}{16}&{}:{}&\frac{3\sqrt{26}+\sqrt{13}+3\sqrt{2}+23}{10}&{}:{}&-\frac{13\left(3\sqrt{26}+\sqrt{13}+3\sqrt{2}+13\right)}{80}&,\\C^\prime&={}&\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13}{24}&{}:{}&-\frac{3\sqrt{26}-\sqrt{13}-3\sqrt{2}+13}{10}&{}:{}&\frac{21\sqrt{26}-7\sqrt{13}-21\sqrt{2}+211}{120}&.\end{alignedat}\]
Approximately,
\[\begin{aligned}\overrightarrow{AA^\prime}&\approx{}-0.055332304304\overrightarrow{AI_A},\\\overrightarrow{BB^\prime}&\approx{}-9.036312625840\overrightarrow{BI_A},\\\overrightarrow{CC^\prime}&\approx{}-3.408144429699\overrightarrow{CI_A}.\end{aligned}\]
\[\begin{alignedat}{4}A^\prime&\approx{}&1.069165380380&{}:{}&-0.033199382582&{}:{}&-0.035965997798&,\\B^\prime&\approx{}&2.259078156460&{}:{}&4.614525050336&{}:{}&-5.873603206796&,\\C^\prime&\approx{}&0.852036107425&{}:{}&-2.044886657820&{}:{}&2.192850550395&.\end{alignedat}\]
6a (231)

Hiroyasu Kamo