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Sometimes beneath a simple problem statement lies unexpected mathematical depth. Consider this challenge from the Connected Geometry curriculum (Education Development Center 2000): "Find as many ways as you can to divide an arbitrary triangle ABC into four equal-area triangles." Before reading further, try solving the problem yourself.
Figure 1 shows a method in which points D, E, and F divide AB(overscored) into equal fourths. This approach leads to three separate solutions, since any of (Delta)ABC's three sides can be evenly quartered. By using this generous counting scheme, in how many ways could you subdivide (Delta)ABC?
If you found fewer than thirty, then you are just scratching the surface. Teachers in my geometry course found one hundred methods for starters, as well as an infinite number when they interpreted the question creatively. This article shows the teachers' work and their flow of ideas from one technique to the next. It also highlights the diverse areas of mathematics that they brought to the problem. The approaches included categorizing, counting, and creating subproblems, with references to congruence, centroids, and dissections.
DIVIDE AND CONQUER
Nearly all my students first used the quartering technique of figure 1. A technique that I call the divide-and-conquer approach ranked a close second. Students connected a vertex of (Delta)ABC to the midpoint of the opposite side, thus forming a median. As figure 2 shows, the median divided the triangle into two smaller triangles, (Delta)ACD and (Delta)BCD, both with half the area of (Delta)ABC. These triangles were then divided into equal-area halves, too. For both triangles, a median could be drawn in three ways, thus yielding 3 x 3, or nine, ways to combine the halves of (Delta)ACD and (Delta)BCD....