An equilateral triangle is inscribed in the parabola  $y^2=4ax,$  such that one vertex of this triangle coincides with the vertex of the parabola. Then length of the side of this triangle. 

As shown in the figure, equilateral triangle OPQ is symmetrical about x-axis

Let OP=1
In triangle OMP,
  $OM=OP\cos {30}^0=\frac{\sqrt{3}l}{2}andMP=OP\sin {30}^0=\frac{l}{2}$
 $\therefore P\equiv (\frac{\sqrt{3}l}{2},\frac{l}{2})$
P lies on parabola 
 $\therefore {\left(\frac{l}{2}\right)}^2=4a\left(\frac{\sqrt{3}l}{2}\right)$
 $\therefore l=8\sqrt{3}a\Rightarrow OP=8\sqrt{3}a$