We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside (or on) a polygon $P$ a rational viewing point if all of the angles formed by $p$ together with any two adjacent vertices of $P$ are rational multiples of $\pi$.
Problem 1. Suppose we have a convex polygon $P$, and there exists a rational viewing point in (or on) $P$. Must there exist infinitely many rational viewing points in (or on) $P$ ?
Problem 2. We observe that whenever we have a vanishing sum of roots of unity, with any real coefficients, then we may consider the individual summands as vertices of a polygon $P$, with the origin as a rational viewing point. In this case, are there always other rational viewing points in (or on) $P$ ?