We are asked to show that the midpoints of the sides of any quadrilateral form a parallelogram. In a coordinate proof we can use the Midpoint Formula.
The midpoint between(x_1,y_1)and(x_2,y_2)is
(x_1+x_2/2,y_1+y_2/2)
Let's find the coordinates of the midpoint of the diagonals of quadrilateral PQRS. Let's start with the midpoint of PR.
Point P is the midpoint between A(a_x,a_y) and B(b_x,b_y).
P(a_x+b_x/2,a_y+b_y/2)
Point R is the midpoint between C(c_x,c_y) and D(d_x,d_y).
R(c_x+d_x/2,c_y+d_y/2)
Let's use these coordinates to find the midpoint of PR.
Since the midpoint of PR and QS is the same, the two diagonals of quadrilateral PQRS bisect each other. According to Theorem 6.11, this means that PQRS is a parallelogram.
