Let E and F be the midpoints of sides BC and AD respectively of convex quadrilateral ABCD and O the intersection of the diagonals AC and BD.

Prove that O lies inside quadrilateral ABEF if and only if

       Area(AOB) < Area(COD).

Let A, B, C,... have position vectors a, b, c,... relative to O.

Using ‘x’ to denote a vector product:

e x f     = 0.5(b + c) x 0.5(a + d)
            = 0.25(c x d  -  a x b)                            (since a x c = b x d = 0)
            = 0.25{cd*sin(COD)  -  ab*sin(AOB)} k
            =0.5{area(COD)  -  area(AOB)} k

where k is a unit vector in the direction of  a x b  and  c x d.

If area(COD) > area(AOB) then it follows that e x f is also in the direction of k, which means that the letters EOF describe a triangle in the same sense (i.e. clockwise or anticlockwise) as do AOB and COD, so that O is inside quadrilateral ABEF. If area(COD) < area(AOB)  then e x f is in the direction of -k, and O is inside CDFE.

Posted by Harry on 2010-08-18 19:27:14