Two points a line, three points a circle... Three points triangle inequality, four points Ptolemy-Euler inequality... I was wondering, it's known that:
Two points determine an unique straight line.
Three points are collinear if and only if they satisfy the triangle “equality” $AB + BC = AC$.
Three points determine an unique circle.
Four points are concyclic if and only if they satisfy the Ptolemy-Euler “equality” $AB\cdot CD + BC\cdot AD = AC\cdot BD$.
But what about:
Four points determine an unique $\underline{\qquad\qquad}$.
Five points are con$\underline{\qquad\quad}$ if and only if they satisfy the $\underline{\qquad}$ “equality” $\underline{\qquad\qquad\qquad}$. 
 A: I doubt that a generalization of these facts can be simply stated without using some elementary algebraic geometry. There is indeed a good setting in which this can be done, though, and it is the geometry of the projective plane.
Straight lines are said to be curves of degree $1$, because they correspond to polynomial equations of the form $a x + b y + c = 0$. Curves of degree $2$ are known as conics. In general, given $d(d+3)/2$ points in a "sufficiently general position", there is exactly one curve of degree $d$ passing through them.
For $d = 1$, being in general position means that the two points are not coincident. For $d = 2$, it means that whenever you pick $4$ of the $5$ points, they are not collinear. But for $d \ge 3$ the condition is not easy to state in geometrical terms.
By the way, the circle is just a special conic. Given $5$ points in general position there is always a conic passing through them, but this is not always a circle. The fact that $3$ non-collinear points always determine a unique circle doesn't come directly from the general phenomenon I mentioned.
