Deduce relations between sides of a triangle if $\cos A + \cos C= 2 - 2\cos B$

Question

If the relation between the cosines of the angles of a triangle is given by

$\cos A + \cos C= 2 - 2\cos B$

then what relation do the sides of the triangle follow? That is, are they in Arithmetic, Geometric or Harmonic progression?

I tried using trigonometric addition formulae and then deducing the results. I applied $\ cosx + \ cosy $ formula along with $A + B + C = \pi$ but I could not solve the question.

Please Note: This question is intended to be a multiple choice question.

Answer 1

Let us assume, additionally, that we have a right triangle with hypotenuse $c=1$. Then the given conditions result in the equations $$ b = 2 - 2a, \qquad a^2+b^2=1. $$ Solving these equations we find $$ a=3/5, \qquad b=4/5, \qquad c=1. $$ Thus a right triangle with sides $a,b,c$ in an arithmetic progression satisfies the given conditions.

Note: We have treated this as a multiple choice question. By producing one example of arithmetic progression we excluded the other options.