Find the radius of a circle circumscribed about an isosceles triangle, the lateral side of which is 2√3

Find the radius of a circle circumscribed about an isosceles triangle, the lateral side of which is 2√3, and the apex angle is 60 degrees.

Using the cosine theorem, we find the length of the base of a given isosceles triangle:

(2√3) ^ 2 + (2√3) ^ 2 – 2 * 2√3 * 2√3 * cos (60 °) = 12 + 12 – 2 * 12 * 1/2 = 24 – 12 = 12.

Applying the formula for the area of a triangle on two sides and the angle between them, we find the area S of this triangle:

S = 2√3 * 2√3 * sin (60 °) / 2 = 12 * (√3 / 2) * 1/2 = 12√3 / 4 = 3√3.

Applying the formula for the area of a triangle in terms of the radius R of the circumscribed circle, we find R:

R = 2√3 * 2√3 * 12 / (4 * 3√3) = 144 / (12√3) = 12 / √3 = 12√3 / 3 = 4√3.

Answer: 4√3.