Let ABCD be a convex quadrilateral with BA = BC and DA = DC. Let E and F be the
midpoints of BC and CD respectively, and let BF and DE intersect at G. If the area of CEGF
is 50, what is the area of ABGD?	Pick one.
A) 100	B) 200	C) 300	D) 400	E) 500
Let [XY Z] denote the area of a polygon XY Z.   Observe that BF and DE are two medians of 4BCD, so G is the centroid of 4BCD. Since   the medians of a triangle divide the triangle into six parts of equal area, we have   [CGF] = [CGE] =   1/2   [CEGF] = 25:   It follows that [CBD] = 6 25 = 150 and [BGD] = 2 25 = 50.   Furthermore, since BA = BC, DA = DC, and BD = BD, we have that 4ABD = 4CBD, so   [ABD] = [CBD] = 150. Finally,   [A200GD] = [A200D] + [200GD] = 150 + 50= 
<gadget id="calculator">150 + 50</gadget>
<output>200</output>
200   correct answer 200
<result>B</result>
Let ABCD be a convex quadrilateral with BA = BC and DA = DC. Let E and F be the
midpoints of BC and CD respectively, and let BF and DE intersect at G. If the area of CEGF
is 50, what is the area of ABGD?