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.

independent. For example, the incenter of $\Delta ABC$ lies on the Euler line of $\Delta ABC$ iff $\Delta ABC$ is isosceles. Hence, if we define $P$ to be the incenter and $Q$ the projection of $P$ onto the Euler line, then $P$ and $Q$ coincide for all isosceles triangles. Does the class of isosceles triangles qualify as an independent set? Two elements in this class are in general not obtained from each other by a rotation or a similarity transformation, but I am wondering whether a map between them is consideredsimple. $\endgroup$