Considering the/a definition of a regular polygon from Wiki :
In Euclidean geometry, a regular polygon is a polygon that is equiangular (all angles are equal in measure) and equilateral (all sides have the same length).
, my question is how to prove that the number of symmetry lines is equal to the number of sides/vertices of a regular polygon?
To prove the statement (i.e. my question), i.e. by means of some logical 'discussion' to start from the definition to arrive to the conclusion I think a way that:
A line of symmetry is a line that if you fold the polygon by that line each side will fit exactly on each other. So choose a point on the edge of some polygon and start to move from that point to another point again on the edge of that polygon with the condition that the line bisects the perpendicular line joining two points of the boundary of the polygon. And because there are n distinct point to start these lines (why? I guess but can't prove) so there are n lines of symmetry in a regular n-gon.
I don't feel this as a rigorous proof and I don't know any other way to prove. Any simple detailed explanation would much be appreciated.
Best Answer
I'm not saying you can't refine the proof you attempted, but the first thing that comes to mind when dealing with symmetries of polygons is a group action of the dihedral group on the polygon. In a sense, group theory is the study of symmetry: when a group acts on a set, each group element describes an operation that will rearrange the set in such a way that it looks like it did before, and only the labels have changed. A permutation is precisely a reordering of elements, and Cayley's theorem states that every group is isomorphic to a subgroup of a symmetric group (which is the group of all permutations on a certain number of elements).
I'll use $D_{2n}$ to denote the dihedral group of $2n$ elements. It is generated by two elements: a reflection $R$ (which you can think of as reflecting an $n$-gon across your favorite axis, say the vertical one), and a rotation $r$ (which you can think of as rotating the $n$-gon clockwise). It should be clear that if you reflect across the same axis twice, you're back to where you started (so $R^2 = 1$), and the same is true if you rotate $n$ times in the same direction (so $r^n = 1$). Another relation on these elements is that if you reflect your polygon, and then rotate it clockwise, and then reflect it again, it's the same as rotating it counterclockwise. In fact, the dihedral group can be presented as follows:
$$D_{2n} = <r,R : R^2 = r^n = 1, RrR = r^{-1}>.$$
This is more information than you asked for, but it means that there are only two kinds of symmetries of an $n$-gon: those made of just rotations, and those made of reflections too. Moreover, there are the same number of each, and the number of symmetries with reflections is just the number of lines of symmetry.