The radius of a circle inscribed in an isosceles triangle is 6√ 3. Find the perimeter of the circle.

The radius of the inscribed circle in the triangle is the point of intersection of the bisectors, and since the triangle ABC is equilateral, then the medians and heights of the triangle are also.

The medians of the triangle, at the point of their intersection, are divided by a ratio of 2/1 starting from the apex of the triangle. Then OB = R = 2 * OH = 2 * r = 2 * 6 * √3 = 12 * √3 cm.

Then ВН = OB + OH = 12 * √3 + 6 * √3 = 18 * √3 cm.

The height of an equilateral triangle is equal to: BH = AC * √3 / 2.

AC = 2 * ВН / √3 = 2 * 18 * √3 / √3 = 36 cm.

Then the perimeter of the triangle ABC is: P = 3 * AC = 3 * 36 = 108 cm.

Answer: The perimeter of the triangle is 108 cm.