Repeated Angles in the Plane for Angles with Algebraic Tangents

We construct a set of points with $\Omega(n^2\log n)$ triples determining an angle $\theta$ whenever $\tan(\theta)$ is algebraic over $\mathbb{Q}$, matching the upper bound of Pach and Sharir. This improves upon the original construction, which was optimal only for $\tan(\theta)=a\sqrt{m}/b$ with $a,b,m$ positive integers.