I've been wondering whether there is some sort of fundamental rule (such as n equations are needed to solve for n variables) underlying the seeming need for the definitions of different shapes.
For example, to define a right angled triangle given points in a plane, we need only the right angle - its characteristic - hence we can write an equation using Pythagoras' theorem which defines it:
$$AB^2 + AC^2 = BC^2$$
However, what about the definition of a rectangle? What fundamental properties does a rectangle possess? I tried the idea of right angles for each of the vertices, but it seems that you can also define a square in just 3 equations. For example:
$$AC=BD$$ $$AB=CD$$ $$AB^2 + AC^2 = BC^2$$
As well as:
$$AD=BC$$ $$AD^2 + BA^2 = BD^2$$ $$CD^2 + CB^2 = BD^2$$
What fundamental truth requires three properties to define a rectangle in terms of points in a plane? Is it related to solving a problem in n variables? Can this rule, if any, be extended to shapes in general in arbitrary dimensions?