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My Favorite Geometry Proof


Friday, August 2, 2013, 1 PM - 4:55 PM


Organizer:  Sarah Mabrouk, Framingham State University



This session invites presenters to share their favorite undergraduate geometry proofs. These proofs should be suitable for Euclidean and non-Euclidean geometry courses as well as for courses frequently referred to as “modern” or “higher” geometry but not those related to differential geometry or (low-level) graduate courses. Proofs must be for theorems other than the Pythagorean Theorem. Presenters must do the full proof, discuss how the proof fits into the course, provide information regarding prerequisite topics for the proof, and discuss associated areas with which students have difficulty and how such concerns are addressed so that students understand the proof. Presenters are invited to discuss how they have modified the proof over time as well as to share historical information for “classic” proofs and explorations/demonstrations that they use to help students understand the associated theorem. Abstracts should include the theorem to be proved/discussed as well as brief background information.


1 PM -  1:15 PM

Pizzas, Calzones, and Crusts: Using Symmetry to Slice Up A Circle

Michael Nathanson, Saint Mary's College of California


My love affair with the so-called Pizza Theorem is now entering its third decade and continues to provide new joys and unexpected twists, both for me and my students at both the high school and college levels. The original theorem shows how to fairly divide the area of a circle in half by drawing four coincident lines meeting at an arbitrary point on the interior. I will explain and prove this result using a basic property of perpendicular chords. I will then show a new dissection proof of how to split a circle fairly among many people and indicate how this proof provides new insights into the fair splitting of an annulus, a sphere, and a cantelope. Whatever that is.


The motivation to share this with students is two-fold: To share beautiful mathematical results and ideas and to communicate the process by which instructive proofs spawn new questions for students to ponder.


1:20 PM -  1:35 PM

Heron’s Formula: A Proof Without Words

Daniel E. Otero, Xavier University          presentation


My current favorite theorem in geometry is a ``proof without words'' for Heron's formula. This ancient proposition gives the area of a triangle in the Euclidean plane in terms of the side lengths: a triangle with side lengths a, b, c has area = [s(s-a)(s-b)(s-c)]1/2, where s= ˝(a+b+c) is its semi-perimeter. Traditional proofs involve a good deal of algebra and/or trigonometry, or the evaluation of a matrix determinant. The one presented here is based on the existence of an incircle and a dissection of some simple figures, so it has a more appropriate geometric “feel” to it.


1:40 PM -  1:55 PM

Heron’s Formula for the Area of a Triangle

Diana White, University of Colorado Denver


This beautiful result states that the area of a triangle is the square root of s(s-a)(s-b)(s-c), where s is the semiperimeter (half the sum of the side lengths a, b, and c). It thus provides a formula for the area of a triangle in terms of just the side lengths. As such, it has both practical applicability and mathematical purity. It first appeared in Heron’s Metrica, which was perhaps written around 75 C.E.


Heron’s proof, couched in modern notation and language, requires only basic knowledge of Euclidean geometry, relying heavily on the use of similar triangles. Yet the proof is a classic example of “deep” mathematical arguments that require minimal prerequisite knowledge, reminding students that sophisticated arguments need not require extensive mathematical background.


Additionally, the proof brings out many key ideas in geometry that are often not intuitive to those newer to the study of geometry; for example, inscribing circles in triangles and drawing auxiliary lines.


Moreover, in this proof students can readily get “lost” chasing the symbols in the algebraic steps, seemingly wandering aimlessly through a series of non-motivated equations that they are just following line-by-line. By focusing on the drawing, and pointing to it to track the various steps, students greatly enhances the understanding of the proof. We discuss how the author guides her students to use this approach to gain a deeper understanding of both the proof itself as well as when to use diagrams as a primary focus, and algebraic steps for the formal recordkeeping.


2 PM -  2:15 PM

Spherical Triangle Area and Angle Sum

Jeff Johannes, SUNY Geneseo


In teaching my geometry class, I spend the first month looking forward to deriving spherical triangle area. Before we get there, students regularly ask me if they can assume the sum of the angles is a straight angle, and I just happily anticipate revealing the magic to them. Over the years, I have enjoyed explaining this argument to several different audiences, ranging from a general audience with no background to those with a sophisticated undergraduate training. In this talk we will assume a familiarity with great circles as straight lines on the sphere to derive a formula for the area of spherical triangles and infer from it the range of possible angle sums for triangles on the sphere.


2:20 PM -  2:35 PM

The Angle Sum Theorem for Triangles on the Sphere

Marshall Whittlesey, California State University San Marcos          presentation


The angle sum theorem on the sphere states that the sum of the measures of the angles in a spherical triangle is greater than 180 degrees (p radians). There are a number of ways of proving this theorem. One standard approach involves the use of tetrahedra inside the sphere, and another involves showing that the area of a triangle is proportional to the spherical excess. Here we present a proof which relies on more basic results of intrinsic spherical geometry rather than on the extrinsic properties of 3 dimensional space. The proof resembles a similar standard proof for the angle sum theorem in hyperbolic geometry.


2:40 PM -  2:55 PM

The Existence of the Nine-Point Circle for a Given Triangle

Stephen Andrilli, La Salle University          presentation          GSP Lab


Every triangle has an associated “nine-point circle” which contains the following nine points: the midpoints of the three sides of the triangle, the feet of the three altitudes to the triangle, and the midpoints of the three segments from the vertices of the triangle to the orthocenter of the triangle. (In special cases, some of these points may coincide.) In this talk, a proof of this classic result will be given. The "nine-point circle" for a given triangle can be constructed in a straightforward manner using compass and straightedge. A corresponding lab that the author created using The Geometer's Sketchpad program that constructs the "nine-point circle" for a given triangle will be introduced.


3 PM -  3:15 PM

Ptolemy’s Theorem

Pat Touhey, Misericordia University          presentation


A well known proof of a classic result that is not as well known as it should be to undergraduate students. We then utilize Ptolemy’s Theorem and some basic geometry of the circle as a prelude to some trigonometry.


3:20 PM – 3:35 PM

Mini Panel Discussion: Beneficial Proofs to Include in a Euclidean (Plane) Geometry Course



Stephen Andrilli, La Salle University          comments

Braxton Carrigan, Southern CT State University

Daniel E. Otero, Xavier University

Pat Touhey, Misericordia University

Marshall Whittlesey, California State University San Marcos


Moderator:  Sarah Mabrouk, Framingham State University


3:40 PM – 3:55 PM

Convex Quadrilaterals

Braxton Carrigan, Southern CT State University


Finding a way to constantly review proof techniques and concepts previously covered in class for students who have fallen behind but continually moving forward in Neutral Geometry provides a challenge for any instructor. This talk will highlight how one theorem can be used to revisit common misconceptions while still exploring new concepts. Showing that a quadrilateral ˙ABCD is convex if and only if $\overline{AC} \cap \overline{BD}$ = Ć can introduce the idea of convex quadrilaterals while revisiting the relationship between segments, rays, and lines. As a bonus this theorem can uses symmetry or “without loss of generality” which is often misused by students.


4 PM – 4:15 PM

Quadrature, the Geometric Mean, Hinged Dissections, and the Purpose of Proof

Clark P Wells, Grand Valley State University


Mathematicians will generally agree that proofs are a good thing (why else would we be talking about our favorites?) and that rigor is important. But as educators, what is the purpose of proof? I would argue that in a perfect world a proof should not only verify the truth of a proposition, but should give insight into the proposition itself. A sad fact is that proofs often do not give insight, and worse, they can sometimes seem to students as if they were written to deliberately obscure insight. Sometimes, though, you can have both rigor and insight.


Among my favorite geometry proofs are quadrature proofs, which I typically discuss in our senior capstone course, The Nature of Modern Mathematics. The idea of quadrature is to create (typically by compass and straight-edge construction) a square that is “the same size” as a given geometrical object.


My very favorite geometry proof is the quadrature of the rectangle for several reasons. One is that the side length of the square obtained is the geometric mean of the side lengths of the rectangle, another is that it can be proven using hinged dissections and then animated using GeoGebra, as I will show in my talk, which leads to insight about what quadrature and the geometric mean really are. Furthermore, by taking a theorem due to the ancient Greeks and proving it using modern technology, I can emphasize the connectedness of mathematical ideas across centuries.


4:20 PM – 4:35 PM

A Simple Proof of the Classification of Conics by the Discriminant

Martin E Flashman, Humboldt State University          resources


The discriminant of a conic curve in the real cartesian plane determined by a non-degenerate quadratic equation of the form Ax2 + Bxy +Cy2 +Dx +Ey + F = 0 is given by B2 - 4AC. I will present a simple proof of the classification of the conic (ellipse, parabola, or hyperbola) by the discriminant. Many proofs of this result in calculus and pre-calculus textbooks are developed using planar rotations. The proof I will present (without rotations) is based on elementary concepts of homogeneous coordinates in real projective algebraic geometry. It is suitable for an upper division geometry course that discusses the interaction of synthetic and algebraic geometry.


4:40 PM – 4:55 PM

It’s Not Hyperbole: A Transforming Proof

Thomas Q Sibley, St. John's University          presentation


Nineteenth century geometry transformed our thinking about geometry, mathematics as a whole and how we see the world -- and not just once, but several times. Hyperbolic, projective and transformational geometries each reshaped mathematics. In turn a geometry course building on these topics can open and engage students’ minds. The proof of the theorem characterizing hyperbolic isometries in the Klein model provides a capstone for these three areas of geometry. In addition, this result links this material to spherical geometry and even to Einstein's theory of relativity.


This page was created and is maintained by S. L. Mabrouk, Framingham State University.

This page was last modified on Monday, September 02, 2013.