# Proving coincidence in Euclidean geometry by using finitely many constellations

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.

• 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. May 17 '19 at 9:48
• 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. May 17 '19 at 12:20

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.