What is the definition of a coordinate proof? And what is an example?

Feb 24, 2016

See below

Explanation:

Coordinate proof is an algebraic proof of a geometric theorem. In other words, we use numbers (coordinates) instead of points and lines.

In some cases to prove a theorem algebraically, using coordinates, is easier than to come up with logical proof using theorems of geometry.

For example, let's prove using the coordinate method the Midline Theorem that states:
Midpoints of sides of any quadrilateral form a parallelogram.

Let four points $A \left({x}_{A} , {y}_{A}\right)$, $B \left({x}_{B} , {y}_{B}\right)$, $C \left({x}_{C} , {y}_{C}\right)$ and $D \left({x}_{D} , {y}_{D}\right)$ are vertices of any quadrilateral with coordinates given in parenthesis.

Midpoint $P$ of $A B$ has coordinates
$\left({x}_{P} = \frac{{x}_{A} + {x}_{B}}{2} , {y}_{P} = \frac{{y}_{A} + {y}_{B}}{2}\right)$
Midpoint $Q$ of $A D$ has coordinates
$\left({x}_{Q} = \frac{{x}_{A} + {x}_{D}}{2} , {y}_{Q} = \frac{{y}_{A} + {y}_{D}}{2}\right)$
Midpoint $R$ of $C B$ has coordinates
$\left({x}_{R} = \frac{{x}_{C} + {x}_{B}}{2} , {y}_{R} = \frac{{y}_{C} + {y}_{B}}{2}\right)$
Midpoint $S$ of $C D$ has coordinates
$\left({x}_{S} = \frac{{x}_{C} + {x}_{D}}{2} , {y}_{S} = \frac{{y}_{C} + {y}_{D}}{2}\right)$

Let's prove that $P Q$ is parallel to $R S$. For this, let's calculate the slope of both and compare them.

$P Q$ has a slope
$\frac{{y}_{Q} - {y}_{P}}{{x}_{Q} - {x}_{P}} = \frac{{y}_{A} + {y}_{D} - {y}_{A} - {y}_{B}}{{x}_{A} + {x}_{D} - {x}_{A} - {x}_{B}} =$
$= \frac{{y}_{D} - {y}_{B}}{{x}_{D} - {x}_{B}}$

$R S$ has a slope
$\frac{{y}_{S} - {y}_{R}}{{x}_{S} - {x}_{R}} = \frac{{y}_{C} + {y}_{D} - {y}_{C} - {y}_{B}}{{x}_{C} + {x}_{D} - {x}_{C} - {x}_{B}} =$
$= \frac{{y}_{D} - {y}_{B}}{{x}_{D} - {x}_{B}}$

As we see, the slopes of $P Q$ and $R S$ are the same.
Analogously, slopes of $P R$ and $Q S$ are the same as well.

So, we have proven that opposite sides of quadrilateral $P Q R S$ are parallel to each other. That is a sufficient condition for this object to be a parallelogram.