# A collection of lines drawn between points in a regular 13-gon - how to determine where the points sit relative to each other?

So I have 4 collections of lines drawn between points each making a path. The angles are measured. The problem I am attempting to solve is to determine whether or not each of the collections of points are a valid path between some subset of points on a regular 13-gon. So for instance, for the second shape are there 3 vertices in a regular 13 gon such that drawing a line between them gives a 60 degree angle (within 5 degrees of error, see context). I'm having a hard time figuring this out myself. I know the geometry involved but I'm just having a hard time applying it to this problem.

Some Context

The context of this problem is actually a clue given in the hunt for the secret zoo level of the video game Accounting Plus VR and this was my attempt to solve what these diagrams mean. So because I measured the angles with an actual protractor they have 5 degrees of precision. So if there are valid points with approximate angles that is also a valid answer. I hope this doesn't make the question unanswerable but this is what I have to work with geometry wise. This is effectively an actual "real" problem with imperfect diagrams, rather than an ideal theoretical problem coming out of a book.

And yes the images are the original images I obtained for reference blown up in size. So if one wants to remeasure the angles in case my protractor skill is flawed, they are welcome to.

• The vertices of a regular $n$-gon break the circle into equal arcs of measure $360^\circ/n$, so any angle joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $180^\circ/n$. Therefore, for a $13$-gon, you'd expect the angles in the path to be multiples of $13.845\ldots^\circ$.
– Blue
Jan 13 '19 at 7:28

The vertices of a regular $$n$$-gon break the circle into equal arcs of measure $$360^\circ/n$$, so any angle formed by joining two vertices to the center must be a multiple of that measure. By the Inscribed Angle Theorem, any angle formed by joining two vertices to another vertex must be a multiple of half that angle, aka $$180^\circ/n$$.
Therefore, for a $$13$$-gon, you'd expect the angles in your paths to be multiples of $$13.845\ldots^\circ$$.
Taking your measurements of $$25^\circ$$, $$30^\circ$$, $$40^\circ$$, $$45^\circ$$, $$50^\circ$$, $$60^\circ$$ as accurate, you'd seem to need an $$n$$-gon that allows for inscribed angles that are multiples of $$5^\circ$$; in that case, $$n=36$$. This is close to $$39$$, which is a multiple of $$13$$. Note that $$26$$ is also a multiple of $$13$$ (as well as being the number of letters in the English alphabet), and it allows for inscribed angles that are a multiple of $$6.932\ldots^\circ$$.
In any event, you can use the $$180^\circ/n$$ formula to check the viability of your guesses for $$n$$ and/or the accuracy of your angle measurements.