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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.

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    $\begingroup$ It would be helpful if you could clarify the meaning of the word 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 considered simple. $\endgroup$ May 17 '19 at 9:48
  • $\begingroup$ Thank you, @PhilippLampe, this is indeed not as easy as I thought. A formal criterion for independence needs to be formulated. BTW, as a personal note: The NRW team was really successful this week in Chemnitz. $\endgroup$ May 17 '19 at 12:20
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No, there is no number $m$ such that two constructions agreeing on $m$ independent inputs must agree on all inputs. Consider:

  • Let $P$ be the point on line $AB$, distinct from $A$, such that $CP=CA$.

  • Let $Q$ be the point on line $AB$, distinct from $A$ and $B$, such that $CQ=\max(CA,CB)$.

These are easy to construct, potentially with clauses for degenerate cases, depending on your definition of "construction step". They obviously agree infinitely often when $CA>CB$, and disagree sometimes when $CB>CA$.

So the agreement of constructions would only follow from agreement on different regions of inputs. Here are some facts about polynomials which provide better analogies:

  • If $f$ and $g$ are piecewise polynomial functions of $x$, with the same pieces of definition, and each piece of degree at most $n$, and any overlapping pieces of $f$ and $g$ agreeing on $n+1$ points, then $f=g$ always.

  • If $f$ and $g$ are piecewise polynomial functions of $(x_1,\ldots,x_k)$, with the same pieces of definition, and each piece of degree at most $n$, and any overlapping pieces of $f$ and $g$ agreeing on $(n+1)^k$ points, then $f=g$ always.

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