Advanced Euclidean Geometry
Excursions for Secondary Teachers and Students
1. Edition September 2008
264 Pages, Softcover
Wiley & Sons Ltd
State curriculum standards are mandating more coverage of geometry,
as are the curricula for pre-service mathematics education and
in-service teaching. Yet many secondary teachers know just enough
geometry to stay one chapter ahead of their students! What's
more, most college-level geometry texts don't address their
specific needs.
Advanced Euclidean Geometry fills this void by providing
a thorough review of the essentials of the high school geometry
course and then expanding those concepts to advanced Euclidean
geometry, to give teachers more confidence in guiding student
explorations and questions. The text contains hundreds of
illustrations created in The Geometer's Sketchpad
Dynamic Geometry® software, and it is packaged with a CD-ROM
(for Windows®/Macintosh® formats) containing over 100
interactive sketches using Sketchpad(TM) (assumes that
the user has access to the program).
Introduction.
About the Author.
Chapter 1: Elementary Euclidean Geometry Revisited.
Review of Basic Concepts of Geometry.
Learning from Geometric Fallacies.
Common Nomenclature.
Chapter 2: Concurrency of Lines in a Triangle.
Introduction.
Ceva's Theorem.
Applications of Ceva's Theorem.
The Gergonne Point.
Chapter 3: Collinearity of Points.
Duality.
Menelaus's Theorem.
Applications of Menelaus's Theorem.
Desargues's Theorem.
Pascal's Theorem.
Brianchon's Theorem.
Pappus's Theorem.
The Simson Line.
Radical Axes.
Chapter 4: Some Symmetric Points in a Triangle.
Introduction.
Equiangular Point.
A Property of Equilateral Triangles.
A Minimum Distance Point.
Chapter 5: More Triangle Properties.
Introduction.
Angle Bisectors.
Stewart's Theorem.
Miquel's Theorem.
Medians.
Chapter 6: Quadrilaterals.
Centers of a Quadrilateral.
Cyclic Quadrilaterals.
Ptolemy's Theorem.
Applications of Ptolemy's Theorem.
Chapter 7: Equicircles.
Points of Tangency.
Equiradii.
Chapter 8: The Nine-Point Circle.
Introduction to the Nine-Point Circle.
Altitudes.
The Nine-Point Circle Revisited.
Chapter 9: Triangle Constructions.
Introduction.
Selected Constructions.
Chapter 10: Circle Constructions.
Introduction.
The Problem of Apollonius.
Chapter 11: The Golden Section and Fibonacci
Numbers.
The Golden Ratio.
Fibonacci Numbers.
Lucas Numbers.
Fibonacci Numbers and Lucas Numbers in Geometry.
The Golden Rectangle Revisited.
The Golden Triangle.
Index.