Triangle ABC is equilateral. The inscribed circle of this triangle touches its sides at points M and N.

Triangle ABC is equilateral. The inscribed circle of this triangle touches its sides at points M and N. The arc length of this circle is 1. What is the perimeter of triangle ABC?

1. The circumference of a circle of radius r is calculated by the formula:
C = 2 * π * r.
Thus, the length of the arc MN will be equal to:
MN = C / 3 = (2 * π * r) / 3.
Since, by condition, MN = 1, then we find the length of the radius r:
(2 * π * r) / 3 = 1;
2 * π * r = 3 (proportional);
r = 3 / (2 * π).
2. The radius of a circle inscribed in an equilateral triangle is found by the formula:
r = a√3 / 6,
where a is the length of the side of the triangle.
a√3 / 6 = 3 / (2 * π);
a = (6 * 3) / (√3 * 2 * π) = 9 / (√3 * π) = 9√3 / (3 * π) = 3√3 / π.
3. The perimeter of an equilateral triangle ABC is:
P = 3 * a = 3 * 3√3 / π = 9√3 / π.
Answer: P = 9√3 / π.