Hello, the_rookie07!
Prove analytically that the line joining the midpoints of two sides of a triangle
is parallel to the third side and equal half of its length.
Given any triangle, we can orient it so that one vertex is at the origin
. . and another is on the xaxis. Code:

 B
 o(p,q)
 * *
 * *
 D oo E
 * *
 * *
 A 0 * * * * * * * * o C 
(0,0) (r,0)

We have the vertices: .
The midpoint of is: .
The midpont of is: .
The slope of is: .
The slope of is: .
. . Therefore: .
The length of
The length of
. . Therefore: .