Proof (3) : Mid-point theorem (coordinate Geometry)

Assume A(a1,a2), B(b1,b2) and C(c1,c2) are the three vertex,
D, Mid point of AB = ((a1+b1)/2, (a2+b2)/2)
E, Mid point of BC = ((c1+b1)/2, (c2+b2)/2)
Slope of DE = ((a2+b2-c2-b2)/2)/((a1+b1-c1-b1)/2) = (a2-c2)/(a1-c1) = Slope of AC.
Length of DE = sqrt[((a1+b1)/2-(c1+b1)/2)^2+((a2+b2)/2-(c2+b2)/2)^2]
=sqrt[((a1-c1)/2)^2 + ((a2-c2)/2)^2]
=1/2 * sqrt[(a1-c1)^2+(a2-c2)^2]
=1/2 * Length of AC
Without Loss of gerenality, this theorem is true for any two sides.
Therefore, the results follow and this is called mid-point theorem.

good!