The Asymmetric Propeller Revisited
Gillian Saenz, Christopher A.-L. Jackson, Ryan Crumley · College Mathematics Journal · 2000
In [l] Martin Gardner proved an asymmetric propeller theorem that was originally proposed by Leon Bankoff. He showed that by connecting one vertex from each of three congruent equilateral triangles inscribed in a circle to the center of the circle, one can form a fourth equilateral triangle. This fourth triangle is formed by connecting the remaining vertices with segments as seen in Figure 1. Then the midpoints of these segments are connected to form a triangle. This works regardless of the arrangement of the original triangles. We confirmed Gardner's findings and proceeded to consider his final question: when working with squares, does the same phenomenon occur? We show that the answer is no. Here we give counterex? amples. Using Geometer's Sketchpad, we attempted to show that the propeller theorem also worked for squares and possibly other polygons. In [1, Fig. 6] a midpoint of a side of each of three squares occurred at the vertices of an equilateral triangle. We had the squares meet at a single point by shrinking the equilateral triangle to a point. If the conjecture is false for this case, then it will not hold true if the point is expanded to a triangle. By experimenting with Geometer's Sketchpad we found that even when using squares inscribed in a circle the propeller theorem was not true, see Figure 2. The triangle shown in this figure was not equilateral nor did its angles remain constant when the squares were moved.