True or False: The circumradius of a triangle is twice its inradius if and only if the triangle is equilateral. Let $R$ be the circumradius and $r$ be the inradius. The if part is clear to me. For an equilateral triangle, the circumcentre, the incentre and the centroid are the same point. So, by property of cebntroid $AG:GD=2:1\Rightarrow AG=2GD$. Thus $R=2r$.

But is the converse true? Whether $R=2r$ implies that the triangle must be equilateral ? We know some relations involving circumradius and inradius, like $R=\dfrac{abc}{4\Delta}, r=\dfrac{\Delta}{s}$, where $\Delta$ is the area of the triangle and $s$ is its semi-perimeter i.e. $s=\dfrac{a+b+c}{2}$. But then how to show that the triangle is equilateral if $R=2r$.
I would be thankful if anyone can help me.

R is the circum-radius and r is thew in-radius. In an equilateral triangle, the cenroid also divides the median in 2:1 ratio.
so, R=(2/3)*(1.732/2)*a, where a is the side of the equilateral triangle
so, r=(1/3)*(1.732/2)*a
so, we see R=2r
