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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) $P$ ?

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No, no, and no. We have countable nets of circles (built on each side). If we allow the side to slide away along the sides of a fixed rational angle, only one point on one circle stays fixed and everything else moves homothetically thus avoiding any countable set in the generic position. However, fixing two sides, we get a countable net of intersection points. This formally gives a pentagon as a counterexample but, with some extra effort, you can, probably do a quadrilateral as well... –  fedja Nov 26 '12 at 6:41
    
@Fedja: Three no's ? ;) But thanks! Yeah, your argument also shows that for a triangle we do in fact have a dense subset of rational viewing points, by building sliding circles through each pair of vertices and considering their countable network of intersections. But I would conjecture that for any n-gon, with n larger than 3, (particularly interesting is the case n=4) we will never again have a dense subset of rational viewing points. –  Sinai Robins Nov 26 '12 at 15:11
    
---Three no's ?--- Yeah: one for problem 1, one for problem 2, and one for the implicit question "Can we tweak the original question a bit to get a nice affirmative answer to it?" that you asked by the very act of making this post. :) –  fedja Nov 26 '12 at 15:12
    
Cute.....but I was genuinely curious about any possible constructions that one can make to generate rational viewing points, and you gave one. Though I agree that it sometimes seems as though one is "tweeking" when one asks a second question, but we learn as we go along, and curiosity about various things/constructions keeps building up. –  Sinai Robins Nov 26 '12 at 15:23

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