difficult geometry

Draw a radius to two vertices of the inscribed triangle. This creates an isosceles triangle with two congruent sides of length 4 and a base that coincides with a side of the inscribed equilateral triangle. Draw an altitude from the vertex angle of this new triangle to its base. This divides the isosceles triangle up into two 30-60-90 right triangles. using 30-60-90 triangle ratios we can determine that the side across from the 60 degree angle has a length of 2√3. This is half the length of one side of the equilateral triangle, so one side has length 4√3, and the perimeter is 12√3.

Ans: D