Solving Problems in Geometry

Insights and Strategies for Mathematical Olympiad and Competitions

This new volume of the Mathematical Olympiad Series focuses on the topic of geometry. Basic and advanced theorems commonly seen in Mathematical Olympiad are introduced and illustrated with plenty of examples. Special techniques in solving various types of geometrical problems are also introduced, while the authors elaborate extensively on how to acquire an insight and develop strategies in tackling difficult geometrical problems.

This book is suitable for any reader with elementary geometrical knowledge at the lower secondary level. Each chapter includes sufficient scaffolding and is comprehensive enough for the purpose of self-study. Readers who complete the chapters on the basic theorems and techniques would acquire a good foundation in geometry and may attempt to solve many geometrical problems in various mathematical competitions. Meanwhile, experienced contestants in Mathematical Olympiad competitions will find a large collection of problems pitched at competitions at the international level, with opportunities to practise and sharpen their problem-solving skills in geometry.

Contents:

• Congruent Triangles:

  • Preliminaries
  • Congruent Triangles
  • Circumcenter and Incenter of a Triangle
  • Quadrilaterals
  • Exercises

• Similar Triangles:

  • Area of a Triangle
  • Intercept Theorem
  • Similar Triangles
  • Introduction to Trigonometry
  • Ceva's Theorem and Menelaus' Theorem
  • Exercises

• Circles and Angles:

  • Angles inside a Circle
  • Tangent of a Circle
  • Sine Rule
  • Circumcenter, Incenter and Orthocenter
  • Nine-point Circle
  • Exercises

• Circles and Lines:

  • Circles and Similar Triangles
  • Intersecting Chords Theorem and Tangent Secant Theorem
  • Radical Axis
  • Ptolemy's Theorem
  • Exercises

• Basic Facts and Techniques in Geometry:

  • Basic Facts
  • Basic Techniques
  • Constructing a Diagram
  • Exercises

• Geometry Problems in Competitions:

  • Reverse Engineering
  • Recognizing a Relevant Theorem
  • Unusual and Unused Conditions
  • Seeking Clues from the Diagram
  • Exercises

Readership: Students, educators and general public interested in geometry and topology.
Key Features:

• There are currently very few books on the teaching of geometry in a systematic manner
• This book not only gives the solutions to geometrical problems, but also insights on how to search for clues and develop a strategy in tackling them. A large number of problems used in competitions are illustrated as examples
• The authors are active and experienced in the training of the national team for the International Mathematical Olympiad competitions