Explorations in Geometry

The following sections are included:

• Preface

• Contents

The following sections are included:

• Triangles

• Similar Triangles

• Congruent Triangles

• Quadrilaterals

• Polygons

• Circles and Angles

• Cyclic Quadrilaterals

• Intersecting Chords

• Inversion
  • Reflection in a line
  • Inversion in a circle

The following sections are included:

• Basic Definitions

• Complementary Angles

• Supplementary Angles

• Extensions

• Addition Formulae

• Double Angle Formulae

• Half Angle Formulae

• The Sine Rule

• The Cosine Rule

• The Area of a Triangle

• Inverse Trigonometric Functions
  • The Sine and Cosine Functions
  • The Tangent Function

• De Moivre's Theorem
  • Roots of Unity and Euler's Formula

• Hyperbolic Functions

• Inverse Hyperbolic Functions

The following sections are included:

• Menelaus' Theorem

• Ceva's Theorem

• Particular Concurrency Results
  • Medians
  • Altitudes
  • Perpendicular Bisectors of the Sides
  • Interior Angle Bisectors
  • Exterior Angle Bisectors
  • The Gergonne Point

• The Wallace-Simson Line

• The Nine Point Circle

• The Euler Line

• The Miquel Point

• Joshua Klehr's Theorem and its Extensions
  • First extension
  • Second extension

The following sections are included:

• Semi-perimeter Formulae

• Half Angle Formulae

• In-radius Formulae

• Escribed Circles

The following sections are included:

• Review

• Basic Definitions

• The Parabola

• The Ellipse

• The Hyperbola

• Parametric Equations
  • Parabolas
  • Ellipses
  • Hyperbolas

• General Second Degree Equations

The following sections are included:

• Elementary Constructions
  • Bisection of a line segment
  • Bisection of an angle
  • Drawing a perpendicular to a line
  • Drawing a parallel line
  • Trisection of a line segment

• Regular Polygons
  • Three Sides – equilateral triangle
  • Four Sides – square
  • Five Sides – regular pentagon
  • Seventeen Sides – regular 17–gon
  • Other polygons
  • The Carpenter's “Theorem”
  • The theory of the Carpenter's “Theorem”
  • The Draughtsman's “Theorem”
  • Theory of the Draughtsman's “Theorem”
  • The Master Carpenter's “Theorem”
  • Comparison of the three constructions

• Real Numbers
  • Integral Multiples
  • Rational Multiples
  • Irrational Multiples

Please refer to the full text

The following sections are included:

• Cleavers

• Splitters

• Collinearity

In an article [3] in Mathematics Today (the journal of the Institute of Mathematics and its Applications), Michael Carding (formerly a high school principal in Cheshire, England) described a visit to India. There, he met a thirteen year old female student who just loved mathematics. She posed him a problem…

The following sections are included:

• Problems from Crux Mathematicorum with Mathematical Mayhem

The following sections are included:

• Problems from Crux Mathematicorum with Mathematical Mayhem

The following sections are included:

• Index