Chapter 1
Advanced Euclidean Geometry
1.1 Role of Euclidean Geometry in High-School Mathematics
If only because in one’s “further” studies of mathematics, the results
(i.e., theorems) of Euclidean geometry appear only infrequently, this
subject has come under frequent scrutiny, especially over the past 50
years, and at various stages its very inclusion in a high-school mathe-
matics curriculum has even been challenged. However, as long as we
continue to regard as important the development of logical, deductive
reasoning in high-school students, then Euclidean geometry provides as
effective a vehicle as any in bringing forth this worthy objective.
The lofty position ascribed to deductive reasoning goes back to at
least the Greeks, with Aristotle having laid down the basic foundations
of such reasoning back in the 4th century B.C. At about this time Greek
geometry started to flourish, and reached its zenith with the 13 books
of Euclid. From this point forward, geometry (and arithmetic) was an
obligatory component of one’s education and served as a paradigm for
deductive reasoning.
A well-known (but not wellenoughknown!) anecdote describes for-
mer U.S. president Abraham Lincoln who, as a member of Congress,
had nearly mastered the first six books of Euclid. By his own admis-
sion this was not a statement of any particular passion for geometry,
but that such mastery gave him a decided edge over his counterparts
is dialects and logical discourse.
Lincoln was not the only U.S. president to have given serious thought1