Midpoint trapezium (trapezoid) theorem generalized

You should have noticed that EF ≤ (AD + BC)/2, with equality only holding when ABCD is a trapezium (trapezoid) with AD // BC.

Can you logically explain why (prove that) your conjecture is true?
Hint: Perhaps first consider the special case of a trapezium and try thinking of a possible construction or a transformation before extending to the general case.

Only if stuck, have a look at my paper A Trapezium Theorem Generalized in At Right Angles, Nov 2013, Vol. 2, No. 3, pp. 53-56, 213 KB, PDF. Also read the elegant Alternative proof by my colleague Poobhalan Pillay.

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Created by Michael de Villiers, 23 April 2013.