Rose Mary Zbiek asked her students to investigate midpoint pentagons, the figures obtained by joining the midpoints of a pentagon in the same order. What is the ratio of their areas? Computer experiments using Geometers' Sketchpad initially suggested that this ratio should be constant, but when attempts to prove this conjecture failed, further exploration led to the opposite assertion, namely that the ratio is not constant; it depends on the shape of the pentagon. The object lesson was that technology can lead students to make conjectures, and it can also provide evidence to show that a conjecture is false. [RMZ]
This investigation formed the basis of an example in the Standards, in dialogue form, showing how students might discover preliminary theorems and then go on to consider more general cases, make conjectures, and accumulate evidence to see whether a conjecture is likely to be true or false. [NCTM]
In the Spring semester of 1999 Professor Thomas Banchoff of Brown University presented the midpoint polygons problem to the students in his Fundamental Problems of Geometry course, Math 104. He mentioned Rose Mary Zbiek's article, and challenged his students to go beyond the earlier investigations. The students discussed the midpoint problem on the interactive course webpage, as well as in class, and came up with a number of new results, including an explicit general formula for the area ratio. This paper is based on the results of that discussion, which is included here.
In this paper we solve the area ratio problem for all planar polygons, including non-convex and self-intersecting polygons. Continue on to see the solution to the problem.