Is there a problem of plane geometry whose analytic reformulation gives a polynomial non-solvable by radicals?

Question

This answer explains that any elementary plane geometry problem can be reduced to the existence of a solution of a polynomial system (called the analytic reformulation).

Question: Is there a problem of elementary plane geometry whose analytic reformulation gives a polynomial non-solvable by radicals?

You can translate the search for the root of a polynomial into a geometric problem. — Hagen von Eitzen, Nov 14 '15 at 22:24
@HagenvonEitzen: Nice! What would be the problem of elementary plane geometry translating the search of the root of $P(x) = x^5-x+1$ ? — Sebastien Palcoux, Nov 14 '15 at 22:30

score 3 · Accepted Answer · edited Nov 15 '15 at 04:21

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I take for granted that $x^5-x+1=0$ has Galois group $S_5$ and hence is not solvable by radicals. It can be translated into a geometric problem:

Given $A,B$, find all remaining point so that you obtain a figure where

triangles $ABC$, $ACD$, $ADE$, $AEF$, $AFG$ are similar
$ABCG$ is a parallelogram

Note that identifying $\vec{AB}$ with $1\in\Bbb C$ and $\vec{AC}$ with $x\in\Bbb C$, the first bullit point ensures that $\vec{AG}=x^5$ and the second ensures that $x^5+1=x$.

edited Nov 15 '15 at 04:21

Sebastien Palcoux

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answered Nov 14 '15 at 22:34

Hagen von Eitzen

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Thanks for adding the graphics! – Hagen von Eitzen Nov 15 '15 at 10:16

Is there a problem of plane geometry whose analytic reformulation gives a polynomial non-solvable by radicals?

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