Proof that the middle line of a trapezoid bisects any segment that joins points from its bases

Question

I know that the middle line of a trapezoid is the line that joins the midpoints of the non-parallel sides, and that it is parallel to the bases of the trapezoid.

How do I prove that the middle line of a trapezoid bisects any segment that joins points from its bases?

Answer 1

Hint: Consider using the intercept theorem.

This theorem tells us the following: Since $DG\parallel EI\parallel AH$:

$$\frac{GI}{IH}=\frac{DE}{EA}=1$$