This is a rigid regular heptagon I found on Wikipedia during associated research for my question on [rigid pentagons](https://math.stackexchange.com/q/3604438):
![](https://upload.wikimedia.org/wikipedia/en/thumb/2/2f/Meccano_heptagon.svg/1024px-Meccano_heptagon.svg.png)

The accompanying text reads

> 
> The construction includes two isosceles triangles which hold the rest of [the] bars fixed. The regular heptagon's side $a$, the shorter isosceles triangle side $e$, and the longer isosceles triangle side $d$ satisfy
>  $$7a^2+e^2=4d^2\tag1$$
> 
> 
> 

Here $a,d,e$ are integral, or more generally rational. The Wikipedia text goes on to say that $(1)$ can be derived from the following identity:
$$\sin\frac\pi7-\sin\frac{2\pi}7-\sin\frac{4\pi}7=-\frac{\sqrt7}2\tag2$$
(If $a=1$ then by solving a Pell equation (see e.g. [here](https://math.stackexchange.com/a/555479/357390)) we get $d=\frac t{16}+\frac7t$ for $t\in\mathbb Q$.)

Now it is easy to prove $(2)$ by a minimal polynomial calculation. It is also easy to prove $(1)$ once you determine that the isosceles triangle's height is $\frac{\sqrt7}2a$. But how does $(1)$ follow from $(2)$?

If the association can be shown, I might be able to derive a rigid *regular nonagon* with rational sticks from the following identity.
$$\sin\frac\pi9+\sin\frac{2\pi}9=\sin\frac{4\pi}9$$