Question

If the in-circle of a $△ABC$ passes through the circumcentre, then the vaue of $(\cos A+\cos B+\cos C{)}^2$ is

Answer 1

Distance between circumcentre and incentre =$\sqrt{R^2-2Rr}$
And in question it is given that the distance between then is $r$
$\sqrt{R^2-2Rr}=r\Rightarrow \frac{r}{R}=\sqrt{2}-1\because (\cos A+\cos B+\cos C)=1+\frac{r}{R}\therefore (\cos A+\cos B+\cos C{)}^2=(\sqrt{2}{)}^2=2$