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### 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/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Linear Algebra

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