$\text{Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle}$

$\text{is along the line x+y=3. If R and r be the radii of circum circle and incircele of triangle ABC the R+r is equal to}$

 

$\begin{array}{l}\text{If the triangle is equilateral triangle, all the centers are coincide with the centroid}\\ \text{The distance from centroid to any side is inradius }r\\ \text{Hence }r\text{\hspace{0.17em}\hspace{0.17em} is the distance from origin to the line }x+y=3\\ r=\frac{3}{\sqrt{2}}\\ \text{In an equilateral traingle, the relation between }R\text{\hspace{0.17em} and }r\text{\hspace{0.17em}is }R=2r=\frac{6}{\sqrt{2}}\\ \text{Therefore,\hspace{0.17em}}R+r=\frac{9}{\sqrt{2}}\end{array}$