triangle mid-segment theorem

Theorem.  The segment connecting the midpointsMathworldPlanetmathPlanetmathPlanetmath of any two sides of a triangle is parallelMathworldPlanetmathPlanetmath to the third side and is half as long.


Proof.  In the triangle ABC, let A be the midpoint of AC and B the midpoint of BC.  Using the side-vectors AC and CB as a basis ( of the plane, we calculate the mid-segment AB as a vector:


The last expression indicates that the segment AB is such as asserted.

Corollary (Varignon’s theorem).  If one connects the midpoints of the of a quadrilateralMathworldPlanetmath, one obtains a parallelogramMathworldPlanetmath.

Title triangle mid-segment theorem
Canonical name TriangleMidsegmentTheorem
Date of creation 2013-03-22 17:46:35
Last modified on 2013-03-22 17:46:35
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Theorem
Classification msc 51M04
Classification msc 51M25
Synonym mid-segment theorem
Related topic MutualPositionsOfVectors
Related topic ParallelogramTheorems
Related topic MedianOfTrapezoid
Related topic CommonPointOfTriangleMedians
Related topic Grafix
Related topic SimonStevin
Related topic InterceptTheorem