Centers of Triangles of Fixed Center: William Mueller 1. Introduction There are many popular misconceptions about undergraduate research projects in mathematics: that they are only for advanced students; that they require a tremendous amount of extracurricular time and effort; that they must be guided by an advisor with a vast and detailed knowledge of the subject; that good topics, simultaneously rich and accessible, are difficult to find. The truth is much simpler. The joys and frustrations of mathematical research can be experienced in any subject, at any level. One must only learn to look closely, with an openness to the possibility of discovery. And "looking" is easier than it has ever been before. Fast computers and friendly software allow students to manipulate their intuitions with a naturalness that most research mathematicians only dreamed of during their own undergraduate days. In the new age of electronic Surrealism 2. Motivations The investigation described here was suggested by some initial computer explorations carried out by students in a multivariable calculus class. Although the results of the investigation are of interest in themselves (in fact, despite their provenance in a subject that has been explored for centuries, I believe that the results are essentially new), we also mean to present the process of the investigation, as an example of how naturally an undergraduate research program may develop. The topic of the investigation comes along in the course of other studies and, at least initially, the investigation is an elementary extension of those studies. Careful, systematic observation, however, reveals the topic's beautiful, unexpected complexities. Early on in multivariable calculus, students must be introduced to vectors. Before vectors are used to describe something else, which is itself new to students (such as space curves), it is instructive to use them in a more familiar setting. Force vectors are a traditional choice for this—intuitive if not always familiar—and their applicability is readily accepted. This ready acceptance comes at a price, however. The algebra of force vectors is so elementary that the notation's powerful simplicity is obscured. The advantages of vector notation are much more apparent in another familiar setting: plane geometry. What most undergraduates seem to remember about this subject, from their high school experience, is a tortuous series of theorems and corollaries, and not a great deal of actual geometry. Presented with vector methods, however, the theorems of high school geometry take surprisingly simple form. Expressed elegantly and succinctly by the notation, the geometry itself becomes apparent. The principal idea of vector-based proofs is to replace selected line segments in geometric figures with vectors, and then substitute the unifying notions of scalar multiplication and (vanishing) dot product for the sundry Euclidean propositions on parallel and perpendicular lines, respectively. For example, consider the theorem that says that the medians of a triangle are concurrent. If such theorems are proved at all in high school texts, they are usually treated algebraically, as a demonstration of analytic geometry, or, alternatively, as a kind of grand crescendo at the end of the text, in which all of the previous propositions are allowed to blow their horns. A tribute to the rigors of mathematical reason, in either presentation, but the geometry is usually lost. By vector methods, the proof is more straightforward: THEOREM.
For some scalar
Equating the two expressions for s = t = 2/3. So = (1/3)(P + A + B). By the symmetry of this expression, we see that each pair of medians would lead to the same intersection. ¤CNot only is the concurrence immediately apparent in this proof, but the fact that it occurs at a trisection point of each of the medians seems to come for free. The notational advantages displayed in such proofs are much more impressive than any of the vector demonstrations that, for example, two people can pull a car out of the mud. Usually, in fact, a certain number of students (often the ones with the most painful memories of high school geometry) are impressed enough to want to do more; to see what other theorems can be proved this way. At which point the advisor-to-be smiles gently. 3. Questions and Answers Readers of this Magazine were reminded, in [ Perhaps the most basic question about these centers is this: where do they lie? That is, if one considers triangles of many different shapes and sizes, what is the locus of a particular center? It isn't difficult to demonstrate that, allowed arbitrary rotations and dilations of a triangle (without translation), any of these centers can be made to fall at any point in the plane—a less than marvelous result. Still, the question remains interesting for specific natural classes of triangles, such as those for which one of the centers is held fixed. In this case, the results are indeed quite marvelous. We will consider two such classes: triangles of a fixed incenter and triangles of a fixed circumcenter. These particular classes are chosen because the auxiliary existence of a fixed incircle or circumcircle, respectively, leads to a parametrization of each class that is especially simple. For each of these two classes, we will consider the locus of the remaining three centers listed above.
This class may be parametrized by the two angles
Each point in this domain represents a particular triangle with its incenter at the origin. For purposes of computation, the domain must be treated discretely; but, of course, the mesh size may be varied to obtain arbitrary detail (at the cost of increased computation time). In order to find the other centers of the triangles in Figure 3, it is first necessary to express the coordinates of the triangle's vertices in terms of the parameters It is instructive to plot these vertices as We see that, for equal increments of the parameters, the triangles themselves do not fill the plane uniformly. Technically, the transformations we consider from the Before considering other centers, a good limbering-up exercise is to compute the values of a more familiar function of . This surface, and its contours, are plotted in Figure 5 over the domain of Figure 3.
There is an obvious minimum, which occurs when With this information in hand, we can then ask the computer to step through the domain in Figure 3, plotting the centroid for each of the triangles in the class. The result is spectacular:
The circle shown is the incircle. The complexity of this figure seems to persist at smaller mesh sizes and greater resolutions. Apparently, the centroids fall only within a very specific region of the plane, the boundaries of which are far from obvious. Providing an accurate description of this region, or of the curves of constant
In this case, we parametrize lines whose direction must have a vanishing dot product with the direction of other lines. The resulting system of equations is predictably not pretty, but it may be solved, once again, with the aid of a symbolic processor (followed by the requisite checking and simplification). The results are plotted in Figure 8.
Again, the centers appear to be confined to a region of the plane with unknown boundaries.
The resulting locus is shown in Figure 10.
A discretization of the parameter domain is shown in Figure 12 for
The coordinates of the vertices are: A plot of the areas of these triangles, given by , is shown in Figure 13.
The two equilateral triangles in the domain produce the maximum areas. Notice, as well, the hollow that snakes around the peaks, corresponding to the two
The circumcircle is shown in each plot, without which the two would be indistinguishable (except for scale). Apparently, the centroids fall only within the central third of the circumcircle, and the orthocenters fall within a diameter on either side. Surely, theorems await here.
After much clicking and whirring, the computer produces Figure 17.
We close with the observations of a student of fixed points, who was seen lingering thoughtfully at the International Surrealist Exhibition in London in 1936.
Acknowledgments The author wishes to thank Betty Mayfield, the editor, and the referees for many helpful suggestions. Reference
Footnotes André Breton, Manifesto of Surrealism (1924) |