[anderskaseorg]

Perhaps an answer to one version of your question is that the Tarski–Seidenberg theorem implies that Euclidean geometry is decidable: there exists an algorithm that finds a proof for any theorem of Euclidean geometry. This algorithm, however, is too slow to be practical in general (double exponential time). The proofs it finds definitely don’t correspond one-to-one with the constructions in any reasonable sense.

The compass encodes a constant-radius constraint, and the string encodes a sum-of-distances constraint, but it’s not at all obvious why these two constraints turn out to be the same under a uniform stretching. There are plenty of similar-looking hypotheses that turn out to be false (for example, a curve of constant offset to an ellipse looks a lot like an ellipse but isn’t one).