Question

If in a triangle ABC, the line joining the circumcentre and incentre is parallel to BC, then $\cos B+\cos C$ is equal to

• A. $0$
• B. $1$
• C. $2$
• D. none of these

Answer 1

Answer: (B)

$1$

$\text{In}△OBL,\cos A=\frac{OL}{OB}=\frac{r}{R}$

$\Rightarrow R\cos A=r\Rightarrow R\cos A=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\Rightarrow \cos A=4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\Rightarrow \cos A=\cos A+\cos B+\cos C-1\Rightarrow 0=\cos B+\cos C-1\Rightarrow \cos B+\cos C=1$