Proving the inradius (r) for an equilateral triangle

Question

I am trying to prove the basic properties of an equilateral triangle given in the image. I can do the first three, but am stumped at the fourth one. Any hints?

Answer 1

Hint: We get the equation by summing up the areas: $$\frac{1}{2}ar\cdot 3=\frac{\sqrt{3}}{4}a^2$$.Solve this equation for $$r$$

Answer 2

Hint:

The medians in a triangle intersect at the centroid, which is located on each median at the $2/3$ of the median length from the corresponding vertex.