Consider the triangle $\mathrm{ABC}$ as shown in the diagram. Use similar triangles to show that if $\angle B=2\angle A$ then $b^2=a^2+\mathrm{ac}$.

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.