Advanced Euclidean
Geometry
What is the center of a triangle?
But what if the triangle is not equilateral?
?
Circumcenter
Equally far from the vertices?
P
A
P
I
Points are on the
perpendicular
bisector of a line
B
segment iff they
are equally far
from the
Geometric
Transformations
Definitions
Def: f is a mapping (function) of a set A into a set B if for every
element a of A there exists a unique element b of B that is paired
with a; this pairing is denoted by f(a) = b. The set A is called the
domain of f,
A Communications Network
?
A Communications Network
What are the desirable properties of the switching box?
1. Every two users must be connected at a switch.
2. Every switch must "look alike".
3. The fewest number (but bigger than 1) of switches should
be
Geometric Constructions
Philosophy of Constructions
Constructions using compass and straightedge have
a long history in Euclidean geometry. Their use
reflects the basic axioms of this system. However,
the stipulation that these be the only tools used in a
The
Inversion
Transformation
A non-linear transformation
The transformations of the Euclidean plane that we
have studied so far have all had the property that
lines have been mapped to lines. Transformations
with this property are called linear.
We will n
Non-Euclidean
Geometry
The Parallel Postulate
Non-Euclidean Geometry is not not Euclidean
Geometry. The term is usually applied only to the
special geometries that are obtained by negating the
parallel postulate but keeping the other axioms of
Euclidean G
Arithmetic and Geometry:
Uncomfortable Allies?
Bill Cherowitzo
University of Colorado Denver
MAA Rocky Mountain Sectional Meeting
April 25, 2008
In the beginning.
Only
there was Geometry
^
Euclid XII.2
Circles are to each other as the squares on the diame
What is a proof?
Proofing as a social process, a communication art.
Theoretically, a proof of a mathematical
statement is no different than a logically valid
argument starting with some premises and ending
with the statement you want proved. However, in
t