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?
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,
I'd like to know how to do this too. :/
It is easy to prove the equality for the trapezium but I have not worked out how to prove it for the inequality.
You do know how to do it ...