Two polynomials $f(x)$ and $g(x)$ of degree $n$ are equal if they are equal for $n+1$ different $x$.
Is anything like this true for Euclidean geometry? Say, I have three arbitrary points in the plane (called $A$, $B$, $C$), and I use $k$ construction steps (like adding a circle or a line, or intersecting existing one-dimensional objects) to get a point $P$. From the same three points, I could also do $l$ construction steps getting a point $Q$.
Now I conjecture that $P$ and $Q$ coincide. Is it possible to check this conjecture by looking at a finite number of constellations for $A$, $B$, $C$? These finitely many constellations of course need to be "independent" in the sense that they could not be constructed from each other through rotations or other simple maps.