AoPS Academy
and 
. Prove that
Equality holds when 
Equality holds when 
Equality holds when 
and 
be the parallel sides of a trapezium 
. Let 
and 
be the midpoints of the diagonals 
and 
. If 
and 
, what is the length of 
?
. Choose 
, then 
.
. Easy, 
.
such that 
and it intersects and (
is on 
and 
on 
). Note that 
and 
. If parallel lines make equal intercepts on one transversal, they make equal intercepts on all other transversals too (try to prove this, but due to lack of time I can't). From this, we have 
and 
. By midpoint theorem, 
. Also by the midpoint theorem (on 
), 
. We finalize: ![\[PQ=EQ-EP=\frac{BC}{2}-\frac{AD}{2}=10-8=\boxed{2}~~\blacksquare\]](http://latex.artofproblemsolving.com/b/1/2/b12dd516d493fbe223edf1177a2030daa967a5a4.png)
and 
are the lengths of the two parallel sides of a trapezium. The following two important results follow from the midpoint theorem.
.
.
.
are collinear.
.
.
.
.
.
.
