Number of equations needed to define a rectangle? 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?
 A: Think about 
$$(AC-BD)^2+(AB-CD)^2+(AB^2+AC^2-BC^2)^2=0.$$
A: Note that with  your $$AC=BD, AB=CD,AB^2+AC^2=BC^2$$ or
$$AD=BC,
AD^2+BA^2=BD^2,CD^2+CB^2=BD^2$$
You do not get a square, you only get a rectangle.
You need to modify your equations.
It would be helpful if you use absolute value for the square as well. 
A: While not exactly what the OP wants, I would suggest the coordinate approach. Then the "triangle equation" becomes a circle equation in the form $x^2+y^2=R^2$ which completely defines a circle centered at the point $(0,0)$.
Because of the corners, there's no such nice equation for a square. But we can make some "square-like" shapes in a similar way using the equation:
$$x^{2n}+y^{2n}=R^{2n}, \qquad n>1$$
Here's an illustration for a few $n$, starting with a circle:

"But none of these are squares!" - anyone would say, and be right.
To define a square we can't just use a single equation. We'd have to deal with the absolute value function.
For example, here's a square with the diagonal $D$, rotated by $\pi/4$:
$$|x|+|y|=D$$

We can easily rotate this one right back, and expand it so it fits with the others:

$$|x+y|+|x-y|=a$$

Where $a$ is now the length of the side.

For a rectangle (the edited question) we just need to scale one of the coordinates. For a general rectangle with the sides $a$ and $b$ we have:

$$\left|\frac{x}{a}+\frac{y}{b} \right|+\left|\frac{x}{a}-\frac{y}{b} \right|=1$$


To get rid of the absolute value we can try squaring twice:
$$\left(\frac{x}{a}+\frac{y}{b} \right)^2+\left(\frac{x}{a}-\frac{y}{b} \right)^2+2\left|\frac{x^2}{a^2}-\frac{y^2}{b^2} \right|=1$$
$$\left(\left(\frac{x}{a}+\frac{y}{b} \right)^2+\left(\frac{x}{a}-\frac{y}{b} \right)^2-1\right)^2=4\left(\frac{x^2}{a^2}-\frac{y^2}{b^2} \right)^2$$
Unfortunately, as it often happens with squaring, we get extra solutions (the dashed lines), which don't lie on the original rectangle:

A: The solution of the equation $1-x^2=0$ is two vertical lines, $x=\pm1$. Similarly, $1-y^2=0$ gives the horizontal lines $y=\pm1$. We can combine these to get a set of four lines:
$$(1-x^2)(1-y^2)=0$$
To get a rectangle from this we need only restrict the domain to $|x| \leq 1, |y| \leq 1$. We can do this with square roots:
$$\sqrt{1-x^2}\sqrt{1-y^2}=0$$
A: Various congruence relations area applicable. In general if you know any 3 pieces of information about a triangle, two angles and a side, two sides and an angle, 3 sides.
In Euclidean Geometry-
SAS: Given two sides of a triangle and the angle they make, you can determine the other angles and the third side. Essentially by the cosine rule.
AAS: Given two angles and a side opposite one of those angles, you can determine the third angle and the remaining side.
Hypotenuse leg: Given hypotenuse and a leg of a right triangle, one can determine the angles and the other leg. 
ASS: Ambiguous case. Given two sides and and non-included angle, only two triangles up to congruency can be formed. 
These have implications regarding other polygons as well.
FYI: AAA only guarantees similarity and not congruence in Euclidean Geometry. 
A: The number of equations needed to define a rectangle must equal the number of unknowns needed to define a rectangle. Three points can uniquely define a rectangle, and each point consists of two coordinates, so there are six unknowns, and therefore six equations.
