If you have an equilateral triangle. That triangle is

\(\frac{1}{3}\sqrt{3}\)
explanation in a minute...

I think I remember reading somewhere that the center of the circle will be 2/3 from any of the 3 vertices of triangle. So if an equilateral triangle creates 2 30:60:90 triangles back-to-back, then the height of the triangle will be \(\sqrt{3}\), and the radius should be 2/3 of that length. But the question asks for the relation of the radius to any side of the equilateral triangle. The relationship of the "height" of the equilateral triangle to a side is \(2:\sqrt{3}\). So this would be \(\frac{2}{3}\sqrt{3}:2\) because the radius of the circle to one side of the triangle. This would be the same as dividing \(\frac{2}{3}\sqrt{3}\) by 2. so 2/3 * 1/2 = 1/3...or \(\frac{1}{3}\sqrt{3}\). I'm not sure if this is correct, but it seems logical to me.

I think this is correct. see the following link:
https://www.ajdesigner.com/phptriangle/isosceles_triangle_inscribed_circle_radius_r.php