Vector Analysis (Calculus IV) question

[J3sushooves]

Homework Statement

Prove: the sum of the squares of the sides of any quadrilateral, minus the sum of the squares of the two diagonals, equals four times the square of the distance between the midpoints of the diagonals.

Homework Equations

A(dot)B = |A||B|cos(theta)
I honestly don't think any equations will help, this is more of a thinking question rather than a calculation one, although I imagine that one may be helpful.

The Attempt at a Solution

Well, calling the sides of the quadrilateral vectors A,B,C, and D
Midpoints of diagonals E and F
The requested proof can be stated as:
|A|^2 + |B|^2 + |C|^2 + |D|^2 - |AB|^2 - |CD|^2 = 4|EF|^2

The problem is, I can't really get any further than that. I know it isn't much, but that is as far as I could get. All of the letters represent vectors btw, sorry I couldn't make prettier equations.

Thanks to any assistance!

 

[lanedance]

how about trying to write everything in terms of the 4 side vectors?

eg in your notation
AB = A+B

perhaps a clearer notation is to give the corners the notation
A,B,C,D
then the 4 sides become the vectors
AB, BC, CD, DA

and the diagonals are
AC = AB + BC = - CD - DA
BD = BC + CD = - DA - AB