Let ABCD be a convex quadrilateral, and let M and N be the midpoints of sides AD and BC, respectively. Prove that MN<=(AB+CD)/2. When does equality occur?

Thanks!

EmperioDaZe Dec 16, 2018

#1**0 **

Let's just say that ABCD is a trapezoid. M and N are the midpoints of the non-parallel sides AD and BC, respectively. There is a theorem that states the line MN is the average of the two parallel sides (AB and CD).

If you try a quadrilateral (cannot be trapezoid or parallelogram) with an extremely long side, you will find that MN < (AB+CD)/2.

This equality works for all convex and real quadrilaterals.

Hope this helps,

- PM

PartialMathematician Dec 16, 2018