Averages and Trapezoids

I rediscovered the equation for a trapezoid yesterday evening and became obsessed with it.

All night I thought of this formula and generalized it to a rectangle and a parallelogram. Nestled comfortably within the formula I found the average of two numbers.

It is easy to see that the area of a trapezoid of any kind is just the average of the lengths of its parallel edges multiplied by its height. There is a geometric significance to this fact. For a line segment running parallel to both bases through the exact middle of the shape has this magical average length. 

This can be generalized to any parallel line segments even if they are collinear. Any line segment made from the relative midpoints between the starting and ending points of these two line segments will have a length that is the average of the lengths of the other two. Perhaps one can realize the average of sets of numbers larger than two in geometry this way.