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Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students



Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students

Alfred S. Posamentier

ISBN: 978-0-470-41256-5 June 2008 264 Pages


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™ (assumes that the user has access to the program).



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.


Ceva's Theorem.

Applications of Ceva's Theorem.

The Gergonne Point.

Chapter 3: Collinearity of Points.


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.


Equiangular Point.

A Property of Equilateral Triangles.

A Minimum Distance Point.

Chapter 5: More Triangle Properties.


Angle Bisectors.

Stewart's Theorem.

Miquel's Theorem.


Chapter 6: Quadrilaterals.

Centers of a Quadrilateral.

Cyclic Quadrilaterals.

Ptolemy's Theorem.

Applications of Ptolemy's Theorem.

Chapter 7: Equicircles.

Points of Tangency.


Chapter 8: The Nine-Point Circle.

Introduction to the Nine-Point Circle.


The Nine-Point Circle Revisited.

Chapter 9: Triangle Constructions.


Selected Constructions.

Chapter 10: Circle Constructions.


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.


First chapter includes brief, quick review of the essentials of basic high school geometry.