Two points determine a line and three points determine a circle. How many points determine a square? a 3-4-5 right triangle? your favorite plane curve or figure?

In fact, what do we mean by the question? While a circle is determined by any three points on it, it is not the case that any set of (say) 100 points on a square determines it; all the points could lie on one side, and then there would be infinitely many squares passing through all of them. Our basic question, carefully formulated, is this: given a simple closed plane curve C (or other plane figure), does it contain a finite point set E such that if C′ is any curve similar to C that passes through every point of E, then necessarily C′ = C?