Vector Proof: Parallelogram Diagonals

Date: 01/20/99 at 14:27:50
From: Stephanie
Subject: Vector proofs

Dear Dr. Math,

I have tried for a very long time to solve these proofs but I just 
don't get them. 

1. Given: Parallelogram ABCD
   Prove: the diagonals of a parallelogram bisect each other 
   (Note: I must use the method: Let X be the midpoint of vector AC 
   and show that vector BX = vector XD)

2. Prove that the line joining the midpoints of two sides of a 
   triangle is parallel to the third side and half its length.

Thanks,
Stephanie

Date: 01/20/99 at 20:32:19
From: Doctor Schwa
Subject: Re: Vector proofs

Question 1:

Let ABCD be the vertices of the parallelogram in that order

   A ------- B
    \         \
     \         \
      D ------- C

Then what you know is that the vector AB = DC and the vector AD = BC.
You also know things like AB + BC = AC as vectors. Let's let O be the 
origin somewhere. Then the midpoint of AC is (OA + OC)/2 = OX. What's 
vector BX then? It's OX - OB. And XD = OD - OX. By combining the value 
I found for point X with some of these other things (and other 
relations, like AB = OB - OA = DC = OC - OD ...) you should be able to 
put together a proof.

Question 2:

If the triangle is ABC, then the midpoint of AB is found by the vector

   OM = (OA + OB)/2

and the midpoint of AC is

   OL = (OA + OC)/2

Then the vector from M to L is ML = OL - OM. See what you can do from 
there.

The key point to recognize in both these proofs is that if you're 
going to have a vector, you need a starting and an ending point. It's
convenient to have everything start from the same point. So instead
of "vector A," which doesn't really make sense, say "vector OA." Then 
vectors like AB can be rewritten as OB - OA. When you've referred 
everything to a common standard (the origin) it's much easier to 
combine them.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/