Why do the Midpoints of Quadrilaterals Make a Parallelogram?

Date: 2/7/96 at 22:17:46
From: Anonymous
Subject: Geometry-midpoints of quadrilaterals

Hello!  I teach grades 7-9 and when we hit quadrilaterals we always 
investigate the idea that if you join the midpoints of any quadrilateral
you will always get a parallelogram.  I have asked several teachers
why this is so and have searched through several math books but I 
have come up dry.  There are always one or two students who ask me
why this works and I'm tired of replying "I really don't know."  Can 
you help me in any way?  Thanks, Susan Wilkie
   (Nova Scotia)

Date: 2/8/96 at 6:11:20
From: Doctor Sarah
Subject: Re: Geometry-midpoints of quadrilaterals

Hello there -

You'll find an explanation in our Project of the Month archive at 

   http://mathforum.org/pom/project2.94.html   

The question was about Midpoints and Quadrilaterals: What figure is 
formed when the consecutive midpoints of the sides of a quadrilateral 
are joined? What if the original quadrilateral were a rectangle? A kite? An 
isosceles trapezoid? A square? A rhombus? Other shapes? Explain why you 
think your answer is true.   Here's part of the explanation given in the 
winning solution sent in by Jennifer Burrows and Amanda Taplett of 
Greenwich Academy, Greenwich, CT:

"When you connect the consecutive midpoints of a quadrilateral, another
quadrilateral is formed inside of it. We will call the original figure 
the mother figure and this new figure the daughter figure. To determine 
what the daughter is, you must examine the diagonals of the mother. It 
is possible to use the diagonals of the mother because the diagonals are 
parallel to the sides of the daughter. 

"The type of quadrilateral that is formed can either be a rhombus, a 
rectangle, or a square, but it will always be a parallelogram. This is 
because when the midpoints are connected to form the sides of the 
daughter figure, each side of the mother figure is bisected. Each newly 
formed side will be parallel to a diagonal of the mother. Two of the 
newly formed sides are parallel to the same diagonal and therefore are 
parallel to each other. Along with the other two sides of the daughter 
that are parallel to the other diagonal of the mother, a parallelogram 
is formed."

There's a lot more explanation, and the solution that received honorable 
mention is also worth looking at.

-Doctor Sarah,  The Math Forum