I'm trying to prove the correctness of the following well-known method to divide a circle into N parts (figure from here):
If I call $O$ as the center of the circle, $R$ to its radius and N to the number of divisions, I see that:
$\triangle AOC$ is a rectangle triangle, with $\angle OAC = 60^\circ$, $\angle AOC = 90^\circ$, and $\angle OCA = 30^\circ$. Furthermore, the length $|AC|=2R$ and the length $|AO|=R$.
In $\triangle AED$, the length $|AD|=4\cdot\dfrac RN$.
In $\triangle AEO$, $|EO|=|AO|=R$.
I want to prove that in $\triangle AEO$, $\angle AOE = \dfrac{2\pi}{N}$ and $\angle OEA = \angle OAE = \dfrac{\pi}{2N}(N - 2)$.
I do not see how to progress from this point.
(PS: doing it just for fun, remembering old times of technical draws)
The attempt to prove the construction correct is doomed to failure, because as it turns out, a heptagon is not constructible by straightedge and compass alone. More details can be found at the Wikipedia link. See also constructible polygon.
The construction is merely an approximation, albeit a pretty good one for a heptagon. The actual angle produced by this method is about $51.52$ degrees; it should instead be $51\frac37$ degrees, which is an error of about one part in $600$ or so. Good enough for most drawings, certainly, but not something I'd build to. (The error is about two-thirds of a degree across the entire heptagon.)