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?

Answer 1

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 $x^5+1=x$.