Construct a large square ABCD with AB at the top.
Next construct a smaller square (A'B'C'D') inside ABCD, with any orientation and centre and join the corresponding corners.
This divides the region between the squares into four.
We will name these divisions (and their areas):
(N)orth = ABB'A'
(E)ast = BCC'B'
(S)outh = CDD'C'
(W)est = DAA'D'.
Show that the areas of these regions satisfy the equality
N+S = E+W.
Let X and Y be complex numbers and O the origin. Define the
cross product
X x Y = X"Y  XY" = 4i*Area(OXY)
where X" and Y" are the complex conjugates of X and Y, i is
the square root of minus 1, and triangle OXY is labeled
counterclockwise.
Let A = 0+0i, B = 1+0i, C = 1+1i, and D = 0+1i be the vertices
of the square ABCD.
Let A' = x+yi, B' = u+vi,
C' = B' + i*(B'  A'), and
D' = A' + i*(B'  A') be the vertices
of the square A'B'C'D'.
4i*Area(ABB'A') = (B  A) x (B'  A) +
(B'  A) x (A'  A)
= (B'  A) x (A'  B)
Using MathCad we can show that
4i*[Area(ABB'A')Area(BCC'B')+Area(CDD'C')Area(DAA'D')] =
(B'  A) x (A'  B) 
(C'  B) x (B'  C) +
(D'  C) x (C'  D) 
(A'  D) x (D'  A) = 0
Therefore,
Area(ABB'A') + AREA(CDD'C') = Area(BCC'B') + Area(DAA'D')
Note: For a simple graphical solution see
www.geocities.com/bractals/2sq.jpg
Edited on January 7, 2006, 2:40 pm

Posted by Bractals
on 20060105 14:04:12 