Affine Transformations, Polynomials, and Proportionality

In the articles [1] and [3], standard tools and techniques of calculus are used to establish a variety of proportionality results concerning areas defined by the lines tangent to a cubic curve, and by the lengths of certain arcs of a parabola, where the arcs themselves are determined by an area-proportionality criterion. We demonstrate here that these results can be viewed as consequences of some basic facts about affine transformations in the plane.