Formulation
You should have noticed that EF ≤ (AD + BC)/2, with equality only holding when ABCD is a trapezium (trapezoid) with AD // BC.
Challenge
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.
Proof Solutions
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. Also read the Alternative proof by my colleague Poobhalan Pillay. Recently also found the same generalization of the midpoint trapezium theorem in Chapter 2 of Mathematical Olympiad Treasures by Titu Andreescu and Bogdan Enescu (2010), pp. 37-38.
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Created by Michael de Villiers, 23 April 2013; modified 27 Feb 2025.