Slaying a geometrical 'Monster' : finding the area of a crossed Quadrilateral

INTRODUCTION Varignon's theorem states that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram. The parallelogram thus formed, known as the Varignon parallelogram, has the property that its area is half that of the original quadrilateral. In Figure 1 the Varignon parallelogram EFGH is formed by joining the mid-points of the sides of quadrilateral ABCD. Proving that the midpoints of an arbitrary quadrilateral doing deed form a parallelogram can readily be accomplished using the midpoint theorem. In Figure 2, one of the diagonals of quadrilateral ABCD has been drawn in. Since H and E are the midpoints of the sides of triangle ADB, it follows that HE is parallel to DB. Using a similar observation in triangle BCD we have FG parallel to BD from which it follows that HE is parallel to FG. Using AC, the other diagonal of quadrilateral ABCD, it can similarly be shown that EF is parallel to HG.