Is there any method how to grab into compass, by using just compass and straight edge, side length of a cube, which is inside a given sphere, touching this sphere by corners, but this sphere is given by radius(diameter) as a circle in 2D plane ... sorry for my English, but I hope you understand what I mean.
2026-03-29 11:05:56.1774782356
Compass and straight edge
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(Unlabeled) image constructed here.
The sphere is centred at $O$. Assume $OA=1$. The inner diagonal of the cube is the diameter of the sphere so has length $2$ and the side of the cube has length $\frac{2}{3}\sqrt{3}$.
Triangle $ABC$ is right-angled at $C$. $AB=2$, $BC=1$, hence $AC=\sqrt{3}$.
Lines $BD$ and $EC$ are parallel and are part of a construction of the trisection of $AC$.
Hence, $AF$ = $\frac{2}{3}AC=\frac{2}{3}\sqrt{3}$ as required.