What is a coordinate proof?

A coordinate proof is a proof of a geometric theorem that uses generalized points on the Cartesian plane to make an argument. This method entails assigning variables to the coordinates of one or more points. These variables are used in the midpoint or distance formulas.

A coordinate proof of the triangle mid-segment theorem, for example, implies that the segment connecting the midpoints of the two sides of a triangle is parallel to the third side and precisely half the length. Where there is no loss of generality, it is assumed that one side of the triangle lies on the X axis with one vertex (0, 0) and the other vertex at (a, 0).