Can one divide a convex polygon into two convex polygons of the same area, with parallelly moving a given straight line on a plane? I was wondering if one can divide a convex polygon into two convex polygons of the same area, with parallelly moving a given straight line on a plane.
Drawing some figures to test, it seems like always possible. It's just my intuition, though. Could one prove or disprove this?

WLOG assume the line is perpendicular to some x-axis, which passes through the polygon. 
When the line first intersects the polygon, the right hand area is larger than the left hand area. As the line is sweeped through, the ratio of the right hand area to the left hand area decreases continuosy from $\infty$ to $0$
Thus, it must be 1 at some point by the intermediate value theorem
Also, the two polygons will always be convex simply because all the angles must be less than 180 degrees
