median of trapezoid

The segment connecting the midpointsMathworldPlanetmathPlanetmathPlanetmath of the legs ( of a trapezoid, i.e. the median of the trapezoid, is parallelMathworldPlanetmathPlanetmath to the bases and its length equals the arithmetic meanMathworldPlanetmath 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:


The last expression tells that  EFAB+DCAB  and  |EF|=|AB+DC|2=|AB|+|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