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In $\triangle \mathrm{ABC}$, if a : b : c $=4: 5: 6$, then the ratio of the circumradius of its inradius is
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$16: 7$
$a: b: c=4: 5: 6$
$\begin{aligned} & \mathrm{R}=\frac{a b c}{4 \Delta} ; r=\frac{\Delta}{s} \\ & \frac{\mathrm{R}}{r}=\frac{a b c}{4 \Delta} \times \frac{s}{\Delta}=\frac{a b c \times s}{4 s(s-a)(s-b)(s-c)} \\ & =\frac{4.5 .6}{4\left(\frac{15}{2}-4\right)\left(\frac{15}{2}-5\right)\left(\frac{15}{2}-6\right)}=\frac{16}{7}\end{aligned}$
$\begin{aligned} & \mathrm{R}=\frac{a b c}{4 \Delta} ; r=\frac{\Delta}{s} \\ & \frac{\mathrm{R}}{r}=\frac{a b c}{4 \Delta} \times \frac{s}{\Delta}=\frac{a b c \times s}{4 s(s-a)(s-b)(s-c)} \\ & =\frac{4.5 .6}{4\left(\frac{15}{2}-4\right)\left(\frac{15}{2}-5\right)\left(\frac{15}{2}-6\right)}=\frac{16}{7}\end{aligned}$
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