Solving Problems in Geometry

Insights and Strategies for Mathematical Olympiad and Competitions

Kim Hoo Hang, Haibin Wang;;;

• Publisher: WSPC
• Year: 2017
• ISBN:
  9789814583763
• Topic: Mathematics
• Subtopic: Topology
• Index: Mathematics

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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.

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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

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Readership: Students, educators and general public interested in geometry and topology.
-->Keywords:Problem-solving;Mathematical Olympiad;GeometryReview: 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

Chapter 1

Congruent Triangles

We assume the reader knows the following basic geometric concepts, which we will not define:

• Points, lines, rays, line segments and lengths

• Angles, right angles, acute angles, obtuse angles, parallel lines (//) and perpendicular lines (⊥)

• Triangles, isosceles triangles, equilateral triangles, quadrilaterals, polygons

• Height (altitudes) of a triangle, area of a triangle

• Circles, radii, diameters, chords, arcs, minor arcs and major arcs

1.1 Preliminaries

We assume the reader is familiar with the fundamental results in geometry, especially the following, the illustration of which can be found in any reasonable secondary school textbook.

(1) For any two fixed points, there exists a unique straight line passing through them (and hence, if two straight lines intersect more than once, they must coincide).

(2) For any given straight line ℓ and point P, there exists a unique line passing through P and parallel to ℓ.

(3) Opposing angles are equal to each other. (Refer to the diagram below. ∠1 and ∠2 are opposing angles. We have ∠1 = 180° – ∠3 = ∠2.)

(4) In an isosceles triangle, the angles which correspond to equal sides are equal. (Refer to the diagram below.)

The inverse is also true: if two angles in a triangle are the same, then they correspond to the sides which are equal.

(5) Triangle Inequality: In any triangle ΔABC, AB + BC > AC.

(A straight line segment gives the shortest path between two points.)

(6) If two parallel lines intersect with a third, we have:

• The corresponding angles are the same.

• The alternate angles are the same.

• The interior angles are supplementary (i.e., their sum is 180°). (Refer to the diagrams below.)

Its inverse also holds: equal corresponding angles, equal alternate angles or supplementary interior angles imply parallel lines.

One may use (6) to prove the following well-known results.

Theorem 1.1.1 The sum of the interior angles of a triangle is 180°.

Proof. Refer to the diagram below. Draw a line passing through A which is parallel to BC. We have ∠B = ∠1 and ∠C = ∠2.

Hence, ∠A + ∠B + ∠C = ∠A + ∠1 + ∠2 = 180°.

An immediate and widely applicable corollary is that an exterior angle of a triangle equals the sum of two non-neighboring interior angles. Refer to the diagram below. We have ∠1 = 180° – ∠C = ∠A + ∠B.

It is also widely known that the sum of the interior angles of a quadrilateral is 360°. Notice that a quadrilateral could be divided into two triangles. Refer to the diagram below.

One sees that similar arguments apply to a general n-sid...