Hello, metlx!

We need the Law of Cosines: .In what ratio are the sides of a triangle if one of the angles is 120°

and the sides of the triangle make up a arethmetic sequence?

so we have 3 sides: a = a, b = a +d , c = a +2d . . . . Good!

and one of the angles is 120° = angle between a and a+d (the shortest sides). . Right!

So we have: .

Since , we have: .

. . which simplifies to: .

And we get: .

The shortest side is: .

The next side is: .

The longest side is: .

The ratio of the sides is: .

. . Multiply by

Therefore, the ratio of the sides is: .