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A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

Consider the triangle $ABC$ as shown in the diagram. Use similar triangles to show that if $\angle B = 2 \angle A$ then $b^2=a^2+ac$.
To find integer solutions of this equation, consider the factors of $a(a+c)=b^2$, 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.