if the sides of a triangle are in AP and the greatest angle of the triangle is double the smallest angle the ratio of the sides of the triangle is

Let the side of triangles a, b, c. Also, the side in AP so , a + c = 2b. ---------------1And also by angle relationship We can easily proof that c² = a(a+c) ------------2 from eq 1 and 2 c² = a(2b) c² = 2ab c = √2ab Now ratio becomes a : b : c = a : b : √2ab Dividing by √ab. √a/√b : √b/√a : √2Or √(a/b) : √(b/a) : √2. Now, from eq 2 a + √(2ab) = 2b adding 2b/4 on both side Then after solving we can get √a = (√5b -√b)/√2Therefore √(a/b) = (√5 - 1) /√2 √(b/a) = (√5 +1 )/ 2√2 The side of triangle in the ratio of 2√5 - 2 : √5 + 1 : 4