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In a $\triangle A B C$, if $a: b: c=4: 5: 6$, then the ratio of the radius of its circumcircle to that of its incircle
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The correct answer is:
$16: 7$
We have, $a: b: c=4: 5: 6$
$$
\begin{array}{rlrl}
\Rightarrow & a =4 x, b=5 x, c=6 x \\
\therefore \quad & s =\frac{15 x}{2} \\
\therefore \quad & \frac{R}{r} =\frac{a b c s}{4 \Delta^2}=\frac{a b c}{4(s-a)(s-b)(s-c)} \\
& =\frac{120 x^3}{4 \times \frac{7 x}{2} \times \frac{5 x}{2} \times \frac{3 x}{2}}=\frac{16}{7} \\
\therefore \quad & R: r =16: 7
\end{array}
$$
$$
\begin{array}{rlrl}
\Rightarrow & a =4 x, b=5 x, c=6 x \\
\therefore \quad & s =\frac{15 x}{2} \\
\therefore \quad & \frac{R}{r} =\frac{a b c s}{4 \Delta^2}=\frac{a b c}{4(s-a)(s-b)(s-c)} \\
& =\frac{120 x^3}{4 \times \frac{7 x}{2} \times \frac{5 x}{2} \times \frac{3 x}{2}}=\frac{16}{7} \\
\therefore \quad & R: r =16: 7
\end{array}
$$
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