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)A(xA,yA), B(x_B,y_B)B(xB,yB), C(x_C,y_C)C(xC,yC) and D(x_D,y_D)D(xD,yD) are vertices of any quadrilateral with coordinates given in parenthesis.
Midpoint PP of ABAB has coordinates
(x_P=(x_A+x_B)/2,y_P=(y_A+y_B)/2)(xP=xA+xB2,yP=yA+yB2)
Midpoint QQ of ADAD has coordinates
(x_Q=(x_A+x_D)/2,y_Q=(y_A+y_D)/2)(xQ=xA+xD2,yQ=yA+yD2)
Midpoint RR of CBCB has coordinates
(x_R=(x_C+x_B)/2,y_R=(y_C+y_B)/2)(xR=xC+xB2,yR=yC+yB2)
Midpoint SS of CDCD has coordinates
(x_S=(x_C+x_D)/2,y_S=(y_C+y_D)/2)(xS=xC+xD2,yS=yC+yD2)
Let's prove that PQPQ is parallel to RSRS. For this, let's calculate the slope of both and compare them.
PQPQ 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)=yQ−yPxQ−xP=yA+yD−yA−yBxA+xD−xA−xB=
=(y_D-y_B)/(x_D-x_B)=yD−yBxD−xB
RSRS 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)=yS−yRxS−xR=yC+yD−yC−yBxC+xD−xC−xB=
=(y_D-y_B)/(x_D-x_B)=yD−yBxD−xB
As we see, the slopes of PQPQ and RSRS are the same.
Analogously, slopes of PRPR and QSQS are the same as well.
So, we have proven that opposite sides of quadrilateral PQRSPQRS are parallel to each other. That is a sufficient condition for this object to be a parallelogram.
