Question

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

Answer 1

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(x_A,y_A)`, `B(x_B,y_B)`, `C(x_C,y_C)` and `D(x_D,y_D)` are vertices of any quadrilateral with coordinates given in parenthesis.

Midpoint `P` of `AB` has coordinates
`(x_P=(x_A+x_B)/2,y_P=(y_A+y_B)/2)`
Midpoint `Q` of `AD` has coordinates
`(x_Q=(x_A+x_D)/2,y_Q=(y_A+y_D)/2)`
Midpoint `R` of `CB` has coordinates
`(x_R=(x_C+x_B)/2,y_R=(y_C+y_B)/2)`
Midpoint `S` of `CD` has coordinates
`(x_S=(x_C+x_D)/2,y_S=(y_C+y_D)/2)`

Let's prove that `PQ` is parallel to `RS`. For this, let's calculate the slope of both and compare them.

`PQ` has a slope
`(y_Q-y_P)/(x_Q-x_P)=(y_A+y_D-y_A-y_B)/(x_A+x_D-x_A-x_B)=`
`=(y_D-y_B)/(x_D-x_B)`

`RS` has a slope
`(y_S-y_R)/(x_S-x_R)=(y_C+y_D-y_C-y_B)/(x_C+x_D-x_C-x_B)=`
`=(y_D-y_B)/(x_D-x_B)`

As we see, the slopes of `PQ` and `RS` are the same.
Analogously, slopes of `PR` and `QS` are the same as well.

So, we have proven that opposite sides of quadrilateral `PQRS` are parallel to each other. That is a sufficient condition for this object to be a parallelogram.