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More Favorite Geometry Proofs Saturday, August 9, 2014, 1 PM – 4:20 PM Hilton Portland, Ballroom Level, Galleria I Organizer: Sarah Mabrouk, Framingham State University |
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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 and should be different from
those presented during the MAA MathFest 2013 paper session (see http://www.framingham.edu/~smabrouk/Maa/mathfest2013/
for more information). 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. |
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1 PM - 1:15 PM |
A Proof of Ptolemy’s Theorem via Inversions Deirdre
Longacher Smeltzer, Eastern Mennonite
University Ptolemy's theorem, attributed to second century Greek mathematician
Claudius Ptolemaeus, gives necessary and sufficient
conditions for one to be able to inscribe a given quadrilateral in a
circle. A standard proof involves
using inscribed angles and similar triangles. A more elegant and modern proof
utilizes an inversion in the plane and resulting properties to establish a
generalization of the theorem. |
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1:20 PM - 1:35 PM |
Archimedes’ Twin
Circles in an Arbelos Dan
C. Kemp, South Dakota State University Proposition 5 from Archimedes’ Book of Lemmas was
popularized in 1954 by Leon Bankoff as a surprise
in an arbelos.
An arbelos consists of three mutually
tangent semicircles with diameters on a common line and lying on the same
side of that line. Archimedes asserts that in an arbelos,
the two circles that are tangent to two of the semicircles and the common
tangent of the smaller semicircles are congruent. Archimedes’ synthetic proof, suitable for presentation in a
geometry course, will be given. Archimedes’ proof is historically interesting
because it contains the first known reference (‘... by the properties of
triangles…’) of the altitudes of a triangle being concurrent. Also a modern
proof using analytic geometry will be presented. If time permits, further
discussion of the twin circles of Archimedes will be given. |
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1:40 PM - 1:55 PM |
Euler’s Famous Line: Gateway to The Harmonic 2:1 Centroid
Concurrency Alvin
Swimmer, Arizona State University It was Archimedes of Syracuse (287-212 b.c.)
that first proved that the three medians of every triangle are concurrent at
the centroid, G, which divides each median in the ratio 2:1. Two millenia
later, in 1763, Leonhard Euler, the most prolific mathematician of all time,
discovered the line determined by the orthocenter, O, and the circumcenter C,
of any (non-equilateral) triangle also contains the centroid G which divides
the interval [O,C] in the ratio 2:1. In more recent times, Tom Apostle and Mamikon
Mnatsakian, discovered in 2004, that the incenter B
and the 1-dimensional center of mass $D$ determine a line which contains the
centroid and G divides the interval [B,D] in the same 2:1 ratio. In 2006, it became clear to me that, the 5
lines and intervals mentioned above, associated with each (non-equilateral)
triangle are part of an infinite family of lines all concurrent at G, Each of
these lines is determined by 2 points which determine an interval divided by
G, in the 2:1 ratio. I call this
family The Harmonic 2:1 Centroid Currency. |
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2 PM - 2:15 PM |
Reflections in
Geometry David
Marshall, Monmouth University Our junior level geometry course provides a study of
Euclidean and non-Euclidean geometries, but leaves much of the specific
content up to the instructor, and the variation can be large. My course
emphasizes transformations, starting with the classification of isometries of
the Euclidean plane. The
Three-Reflections Theorem plays a central role in this classification and
helps with several other pedagogical and content goals. It provides students with an opportunity to
(1) seek generalizations later in the course, (2) experiment with a “proof by
generic example”, (3) make good practical use of available technology, and
(4) do a little group theory. The
theorem (and related results) also serves as an entry way into more general
discussions of classification theorems, Klein's Erlanger Program, and the
role of group theory in geometry. |
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2:20 PM - 2:35 PM |
Reflections on Reflections Thomas
Q. Sibley, St. John's University The Common Core State Standards emphasize geometric
transformations, highlighting the Three Mirror Theorem: Every Euclidean plane
isometry is the composition of at most three mirror reflections. The proofs
for this theorem and the theorems leading to it also generalize beautifully
to hyperbolic and spherical geometries.
Even more, they generalize to higher dimensions for all three of these
geometries with only minor modifications.
Future secondary teachers, indeed all mathematics majors, can gain
important insight by considering these theorems from this unifying
perspective. |
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2:40 PM - 2:55 PM |
The Shortest Path
Between Two Points and a Line Justin
Allen Brown, Olivet Nazarene University In a geometry classroom, asking the question in the
open-ended form above often leads to interesting ideas from students. Many of their ideas do not yield the
shortest path in general, but result in an interesting discussion. And once students realize that their
initial idea is incorrect, they are invested in finding or at least seeing
the correct proof. We will discuss
some of these incorrect ideas, as well as the proof of the theorem, which
uses similar triangles. |
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3 PM - 3:15 PM |
The Perfect Heptagon from the Square Hyperbola It is well known that the perfect 7-sided heptagon cannot be
constructed with a compass and a straight edge alone, but it can be done with
an angle trisector. However, virtually all perfect heptagon constructions
include a “magic” step, which presents some angle 3q, followed immediately by presenting the angle q, then continuing on with the construction. It is also well known that angle trisection
cannot be done with a compass and a straight edge alone because it requires
solving an irreducible cubic, but that it can be done with the help of a
non-circular conic section. The first
conic section we usually learn about is the inverse relationship |
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3:20 PM – 3:35 PM |
The Many Shapes of
Hyperbolas in Taxicab Geometry Ruth
I. Berger, Luther College Taxicab geometry is a good topic for open-ended explorations
in non-Euclidean geometry. My undergraduate students are always surprised at
the many different looking hyperbolas they discover. A simple geometric argument can be given to
classify the different hyperbola shapes, as well as the other conic
sections. The shape depends on the
slope of the line connecting the foci.
The underlying reason for this is that in taxicab geometry circles
have sides of slope 1 and -1. |
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3:40 PM – 3:55 PM |
Geometry Knows
Topology: The Gauss-Bonnet Theorem Jeff
Johannes, SUNY Geneseo Last year I spoke in this
session about angle sum of spherical triangles. And I think now “I cannot
have two favorite geometry theorems.”
On the other hand, what if we think of geometry in a different
context? What if we work with surfaces of non-constant curvature for a
change? Now curvature is a variable
quantity. We cannot speak any longer of the simple situation of the sphere, but
instead we talk about a more sophisticated view of changing geometry. In this context, we cannot ask what the
curvature of the whole surface is, but we can consider the integral of the
curvature over the entire surface.
From this perspective we can zoom out from those triangles with their angle
sums and see global topological properties of our surface. In this talk we will move from angle sum
through a sequence of generalizations and new ideas to find the famous and
powerful Gauss-Bonnet Theorem. |
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4 PM – 4:15 PM |
Finding the Fermat Point by Physics and by Transformation Philip
Todd, Saltire Software My new favorite way of finding the point with minimum sum of
distances to the three vertices of a triangle involves a mechanical thought
computer described in Mark Levi’s “The Mathematical Mechanic”. An interactive model of this mechanical
system can motivate the conjecture that for some triangles the solution lies
at the point which subtends equal angles to the sides. It can also motivate accurate conjectures
on conditions for this solution to apply.
An alternative exploration tool for students with no physics
background will also be presented.
Again, this will allow for both the solution and the conditions under
which the solution applies to be induced. When minimizing the length of a path, I like proofs which
apply transforms to create a path equal to the one to be minimized, but where
the minimal path is clearly a straight line.
We can use reflections in this way to solve Fagnano’s problem of
finding the inscribed triangle with minimal.
Can we apply transforms to reduce the Fermat Toricelli
Point problem to finding the shortest distance between two points? Our conjecture suggests that 120 degree
rotations may be the transforms of choice.
Application of these rotations yields a visual proof both of the
result for triangles which have no vertices over 120 degrees, and of its
breakdown when there are angles greater than 120 degrees. |
