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 

This
session invites presenters to share their favorite undergraduate geometry
proofs. These proofs should be suitable for Euclidean and nonEuclidean
geometry courses as well as for courses frequently referred to as
"modern" or "higher" geometry but not those related to
differential geometry or (lowlevel) 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. 
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. 
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. 
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 (287212 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 (nonequilateral) 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 1dimensional 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 (nonequilateral) 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. 
2 PM  2:15 PM 
Reflections in
Geometry David
Marshall, Monmouth University Our
junior level geometry course provides a study of Euclidean and nonEuclidean
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 ThreeReflections 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. 
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. 
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 openended 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. 
3 PM  3:15 PM 
The Perfect Heptagon from the Square Hyperbola It is well known that the perfect 7sided 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
noncircular conic section. The first
conic section we usually learn about is the inverse relationship _{} which makes it a
nice item to use for implementing a trisection construction within a heptagon
construction. Starting with an unmarked xycoordinate
system and a single square hyperbola H, we show that adding 4 circles and 10 lines gives us the _{} angle for the
perfect heptagon. The proof of why it works and the motivation for how a
square hyperbola naturally arises provide a nice example of combining both
constructive geometry and elementary trigonometry. 
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 openended explorations
in nonEuclidean 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. 
3:40 PM – 3:55 PM 
Geometry
Knows Topology: The GaussBonnet 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 nonconstant 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 GaussBonnet Theorem. 
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. 
This
page was created and is maintained by S. L. Mabrouk, Framingham State
University.
This
page was last modified on Wednesday, June 25, 2014.