Question

If in $△ABC$, $a\cos B\cos C+b\cos C\cos A+c\cos A\cos B=\frac{m△}{R}$.Find $m$

Answer 1

ABC is a triangle.

$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R}$-------1

$\Rightarrow △=2R^2\sin A\sin B\sin C$

$\Rightarrow \frac{m△}{R}=2Rm\sin A\sin B\sin C$

Given,

$a\cos B\cos C+b\cos A\cos C+c\cos A\cos B=\frac{m△}{R}$

$\Rightarrow \cos B(a\cos C+c\cos A)+b\cos A\cos C=\frac{m△}{R}$

$\Rightarrow b\cos B+b\cos A\cos C=\frac{m△}{R}$ ( As $b=a\cos C+c\cos A$).

$\Rightarrow b(-\cos (A+C)+\cos A\cos C)=\frac{m△}{R}$ ( As $A+C=\pi -B$)

$\Rightarrow b\sin A\sin C=\frac{m△}{R}$

$\Rightarrow 2R\sin A\sin B\sin C=\frac{m△}{R}$ (As $b=2R\sin B$)

$\Rightarrow 2R\sin A\sin B\sin C=2Rm\sin A\sin B\sin C$

$\therefore m=1$