Consider the triangle ABC as shown in the diagram. Use similar triangles to show that if ∠B = 2 ∠A then b²=a²+ac.

To find integer solutions of this equation, consider the factors of a(a+c)=b², and that a and a+c have no common factors, so a and a+c must be perfect squares. This will lead to a parametric representation of a, b and c in terms of two parameters and you can use this to generate the triples.