Given any convex polygon in the plane, is it always possible to find a point $p$ in its interior such that when we draw the line segments from $p$ to each of its vertices, the angles formed at $p$ are all (not necessarily equal) rational multiples of $\pi$? 


For a triangle $T$, it's easy to construct such a point, namely the Steiner point $p$ will do, enjoying three angles of measure $2\pi/3$ each, between $p$ and any two adjacent vertices of $T$. But is this known in general?



Seeing the vertices of a polygon with rational angles
Consider the manifold of all $k$-sided polygons. This is $2k-3$ dimensional. Given a set of angles $\theta\_1,...,\theta\_k$, the manifold of polygons with those angles from a point is $k$-dimensional, since it's determined by the distances of the vertices from the point. 


You're not going to cover a $2k-3$ - dimensional manifold with countably many $k$-dimensional manifolds if $k\geq 4$. In particular, these maps are real algebraic maps, and thus aren't anything like space-filling curves.


