Question

Prove that $\frac{r_1+r_2}{1+\cos C}=2R$

Answer 1

$r_1+r_2=4R(\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}+\sin \frac{B}{2}\cos \frac{A}{2}\cos \frac{C}{2})$
$=4R\left(\cos \frac{c}{2}(\sin \frac{A}{2}\cos \frac{B}{2}+\sin \frac{B}{2}\cos \frac{A}{2})\right)$

$=4R\left({\cos }^2\frac{c}{2}\right)=2R(1+\cos C)$
$\therefore \frac{r_1+r_2}{1+\cos C}=2R$.

Hence proved.