Im am studying the book Vanden Cirkel from Ludolph van Ceulen in 1600. In a chapter on side lengths of inscribed $n$-gons. He postulates the following:
Which roughly translates to: I foud the side lenght of the inscribed $55$-gon in a circle of radius $1$ trough the inscribed $10$-gon and $11$-gon.
Now I see where why we could use the side lengt of the $11$-gon. We could use that the length of the inscribed $55$-gon. By using the following:
Let $x$ be the side lengt of the inscribed $55$-gon then we find that $x$ satisfies $$5x-5x^3+x^5=\text{side lenght 11-gon}$$ In particular the smallest positive solution to this equation. But I don't see how we can use the inscribed $10$-gon.
If $(A_1,\ldots A_{10})$ are the vertices of 10-gon, $(B_1,\ldots B_{11})$ are the vertices of 11-gon, and $A_1\equiv B_1$, then $A_3B_3$ is the side of 55-gon.