The radius of a circle inscribed in a regular triangle is √3. Find the side of this triangle.

Let us denote by x the side of this regular triangle.
We use the formula for the area of ​​a triangle through the radius of the inscribed circle S = p * r, where S is the area of ​​the triangle, p is the half-perimeter of the triangle, and r is the radius of the circle inscribed in the triangle.
According to the condition of the problem,
r = √3.
Since the side of this regular triangle is x, the semiperimeter of this triangle is:
p = 3 * x / 2.
Since the angles in a regular triangle are equal to π / 3, the area of ​​this triangle is equal to
S = (1/2) * x ^ 2 * sin (π / 3) = (1/2) * x ^ 2 * (√3 / 2) = (√3 / 4) * x ^ 2 =
Substituting these values ​​of r, p and S in the formula S = p * r, we get:
(√3 / 4) * x ^ 2 = (3 * x / 2) * √3.
We solve the resulting equation:
(√3 / 4) * x ^ 2 – (3 * x / 2) * √3 = 0;
(√3 / 4) * x ^ 2 – (3 * √3 / 2) x = 0;
√3 / 2 * x * (x / 2 – 3) = 0:
x / 2 – 3 = 0;
x / 2 = 3;
x = 6.
Answer: The side of this regular triangle is 6.