A Review of Geometry

Khor Shi-Jie

April 18, 2012

Contents

1 What’s Next? 2

1

Chapter 1

What’s Next?

I have learnt all theorems and properties in planar geometry. What’s next for me?

Similar to the topic of inequalities in algebra, geometry is a topic in Mathematics Olympiad whichis saturated with theorems and properties. What is even more challenging is that students must beequipped with strong visualisation skills in order to incise geometric problems with the correct theoremsand properties. This reason alone makes many student detract attention from geometry and focus onother ares instead.

Yet the mastery of geometry is not an unthinkable feat. Just like what I said in the previous doc-ument on Algebra, there are two distinctive phase in the grasping of Mathematics Olympiad. The firststep in studying geometry is, obviously, to study geometric theorems and be well acquainted with basicapplications of such theorems. Of course, this step is relatively challenging in studying geometry due tothe sheer number of different quantities, shapes and concepts that students have to master. On the otherhand, it is no less important for student to learn the strategies involved in solving geometric problems.It is a cardinal sin (well, pardon my exaggeration) to neglect this step because most geometric problemsin competitions does not merely involve the application of a theorem. It is often coupled with ingenioustransformations and inspired combination of theorems that allows one to solve a geometric problembeautifully. In order to acquire different skills in geometry, it is important for one to do hundreds andthousands of problems to familiarise oneself with different strategies and pick up useful lemmas in theprocess. (You’ve heard it, THOUSANDS)

There are two approaches in solving geometric problems: synthetic geometry and analytical geometry.Synthetic geometry involves using theorems learnt in MO to solve a geometric problem. This is oftencoupled with creative construction and transformation in the process of solving the problem. The lack ofa fixed approach or algorithm in solving the problem is a huge contrast to analytical geometry. Analyticalgeometry, on the other hand, is also known as the brute force method. Students can use methods suchas coordinate geometry, trigonometry, vectors and complex numbers to quantify everything within theproblem and solve the problem algebraically. I do not have any preferred method in solving geometricproblems. Synthetic geometry does help in solving a problem quickly and elegantly, but sometimes ittakes too much time in arriving at the correct approach. The analytical geometry approach will definitelyinvolve heavy computation, but it is more certain that one will obtain the solution through this method.I recommend students to master both techniques in solving geometric problems.

By saying basic theorems and properties in geometry, I refer to the following topics:

1. Planar Geometry

(a) Triangles

i. Similarities and congruences

ii. Area of triangles

iii. Trigonometric properties

iv. Special lines within the triangle and notable theorems (Pythagorean theorem, Mid-pointtheorem, Angle-bisector theorem, Stewart’s theorem, etc)

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CHAPTER 1. WHAT’S NEXT? 3

v. Centres of a triangle

(b) Circles

i. Basic circular properties

ii. Cyclic quadrilaterial

iii. Power of a point and radical axis

iv. Notable theorems (Ptolemy’s theorem, Simson’s theorem, etc)

(c) Concyclic, collinearity, concurrency

i. Menalaus’ Theorem and Ceva’s Theorem

(d) Area properties

i. Heron’s formula

ii. Area relations with circumradius and inradius

iii. Area relations with trigonometry

(e) Geometric inequalities

i. Triangle inequalities

ii. Other geometric inequalities

2. Trigonometry

(a) Trigonometric identities

(b) Sine rule and cosine rule

3. Analytical geometry

(a) Coordinate geometry

(b) Vectors

(c) Complex numbers

(d) Inversive geometry

Yup, that’s a lot of things to study, considering that this is merely the first step in studying geome-try. Note that I have omitted 3D geometry in the list above. 3D geometry is covered in most regionalOlympiads around the world. However, it is not within the syllabus of SMO and IMO. Nevertheless, itis still useful to learn the skills in 3D geometry as these problems do appear in AMC, AIME and PurpleComet.

The fundamental knowledge listed above will help in the following three strategies that I will expoundin the following chapters:

1. Chasing. This technique involves using geometric theorems to ”chase” all the quantities within thegeometric diagram. This includes the lengths, angles and areas within the problem. Similarity andcongruences help tremendously in this technique. Often, we cannot chase all the quantities andhave to set variables to simplify our work (be cautious not to complicate it).

2. Construction and transformation. This technique involves constructing auxiliary lines (or cir-cles/curves) to help us identify similarities, chase quantities or simply apply geometric theorems. Wecan also transform a certain part of the diagram through reflection, rotation, translation, dilation,etc. There are often hints in the question that suggests us to use this technique.

3. Analytical geometry. Brute force method. Nuff said.

I have organised this set of notes based on the strategies used in solving geometric problems. Thisset of notes is created to help students in senior section (and perhaps some in junior section), so I willtry my best in choosing simple examples...