If the lengths of the sides of a triangle are in A.P and the greatest angle is double the smallest, then a ratio of lengths of the sides of this triangle is:

The sides of the triangle are in Arithmetic Progression (AP), so let's denote them as a-d, a, a+d.

According to the law of cosines, we have:

cosC = (a² + (a+d)² - (a-d)²) / 2a(a+d) cosA = ((a+d)² + (a-d)² - a²) / 2*(a+d)*(a-d)

Given that the greatest angle is double the smallest, we have:

cosC = 2cosA

Substituting the values of cosC and cosA from the above equations, we get:

(a² + (a+d)² - (a-d)²) / 2a(a+d) = 2*((a+d)² + (a-d)² - a²) / 2*(a+d)*(a-d)

Solving this equation, we get:

a² - d² = 4d²

This simplifies to:

a² = 5d²

Taking the square root of both sides, we get:

a = d√5

So, the ratio of the lengths of the sides of the triangle is:

a-d : a : a+d = √5 - 1 : √5 : √5 + 1