median of trapezoid

The segment connecting the midpoints of the legs (http://planetmath.org/Trapezoid) of a trapezoid, i.e. the median of the trapezoid, is parallel to the bases and its length equals the arithmetic mean of the legs.

Proof.  Let $AB$ and $CD$ be the bases of a trapezoid $ABCD$ and $E$ the midpoint of the leg $AD$ and $F$ the midpoint of the leg $BC$.  Then the median $EF$ may be determined as vector as follows:

$\overrightarrow{EF}$ $=\overrightarrow{ED}+\overrightarrow{DC}+\overrightarrow{CF}$

$={\displaystyle \frac{1}{2}}\overrightarrow{AD}+\overrightarrow{DC}+{\displaystyle \frac{1}{2}}\overrightarrow{CB}$

$={\displaystyle \frac{1}{2}}(\overrightarrow{AD}+\overrightarrow{DC}+\overrightarrow{CB})+{\displaystyle \frac{1}{2}}\overrightarrow{DC}$

$={\displaystyle \frac{1}{2}}\overrightarrow{AB}+{\displaystyle \frac{1}{2}}\overrightarrow{DC}$

$={\displaystyle \frac{1}{2}}(\overrightarrow{AB}+\overrightarrow{DC})$

The last expression tells that  $\overrightarrow{EF}\parallel \overrightarrow{AB}+\overrightarrow{DC}\parallel \overrightarrow{AB}$  and  $|\overrightarrow{EF}|={\displaystyle \frac{|\overrightarrow{AB}+\overrightarrow{DC}|}{2}}={\displaystyle \frac{|\overrightarrow{AB}|+|\overrightarrow{DC}|}{2}}$.  Q.E.D.

Title	median of trapezoid
Canonical name	MedianOfTrapezoid
Date of creation	2013-03-22 17:46:44
Last modified on	2013-03-22 17:46:44
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	6
Author	pahio (2872)
Entry type	Theorem
Classification	msc 51M04
Classification	msc 51M25
Related topic	MutualPositionsOfVectors
Related topic	MidSegmentTheorem
Related topic	TriangleMidSegmentTheorem
Related topic	HarmonicMeanInTrapezoid