Advanced Euclidean Geometry

Contents

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
• Desarguess 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