Question

In a triangle $ABC$, the line joining the circumcentre to the incentre is parallel to $BC$, then evaluate $\cos B+\cos C$

Answer 1

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

$\Rightarrow R\cos A=r$

$\Rightarrow R\cos A=4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$\Rightarrow \cos A=4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

$\Rightarrow \cos A=\cos A+\cos B+\cos C-1$

$\Rightarrow 0=\cos B+\cos C-1\Rightarrow \cos B+\cos C=1$