A small square is placed inside a big square. The vertices of the small square are joined to vertices of the large square so as to divide the region between the squares into four quadrilaterals, with areas, in order, a, b, c, d.
Prove that a+c=b+d.
The following checks out with Geometer's Sketchpad:
[ABXW] + [CDZY] = [BCYX] + [DAWZ]
(ABxAX + AXxAW) + (CDxCZ + CZxCY)
= (BCxBY + BYxBX) + (DAxDW + DWxDZ)
AXxBW + CZxDY = BYxCX + DWxAZ
Where PQxRS denotes the cross product of
vector PQ with vector RS.
Square WXYZ with respect to square ABCD
has no restriction with respect to size,
center, or rotation.
Edited on November 18, 2012, 7:05 pm
Edited on November 18, 2012, 7:26 pm
Posted by Bractals
on 2012-11-18 19:02:57