4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point  lies on the circle centered at the origin and containing the point (0, 2).

Students will need to know several major formulas and equations in order to prove they can hold their own against Descartes. Here are a few of the most important equations:

• the distance formula: 
 

 
• the slope formula: 
 

 
• the midpoint formula: 
 
• the equation of a circle: (x – h)² + (y – k)² = r²
 

 
• the equation of an ellipse: 
 
• the equations of all the rest of the shapes/functions that can be plotted on the coordinate plane (parabolas, hyperbolas, etc.)
 

They'll also need to be pretty fluent in the properties of geometric shapes. If they don't know that the sum of any two sides of a triangle is always greater than the remaining side, or that all four sides of a rhombus are congruent, they may or may not be in some serious trouble.

To prove a geometric theorem about a shape on the coordinate plane, it's not enough for students to know these properties and formulas. They'll need to be able to apply them to the coordinate plane as well.

Most statements students need to prove will somehow involve angles and lengths. For instance, to prove that a given figure is a square, students could prove that the slopes of adjacent sides are perpendicular (slope formula), and that all sides are the exact same length (distance formula).

There are infinite (well, it might seem that way) possibilities for algebraically proving the many properties of the many shapes in the coordinate plane. While students don't have to know all the ways to prove a particular geometric theorem, they should be able to come up with at least one.