First of all, just to clarify when I say an equilateral pentagon what I mean is a pentagon whose sides are all the same length (not necessarily equal sized angles).
I am going through Euclid and Beyond by Harthstone and at the top of page $147$ there is the following question:
For clarity, when the book says the associated plane $\Pi$ it means the Cartesian plane over the ordered field $F$ (the length of a line segment is defined the usual way via Pythagoras, even if the length itself is not an element of $F$).
Part (a) is straightforward (just compare the ratio of lengths between the triangle's height and side), but part (b) is really stumping me. $\mathbb{Q}(\sqrt{3})$ is easy enough (place an equilateral triangle on top of a square), but I'm not sure about the others or the disproof for $\mathbb{Q}$.
It is clear that the gradient of any line segment must lie in $F$, and similarly $\tan(\alpha) \in F$ for any interior angle $\alpha$ of the pentagon. But this doesn't seem very helpful.
Ideally I would like to have a proof that is more than trial and error or brute force, which can be generalized to other fields (and perhaps to equilateral polygons with an arbitrary number of sides). Can anyone help point me in the right direction?