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Euclid’sGardenA Journey Through GeometryAnand PrakashLevels 3 to 7

Euclid’s Garden

A Journey Through Geometry

By Anand Prakash, who makes Kiwimath.

For
Levels 3 to 7, from Class 5-6 up to INMO preparation
Subject
Geometry
Inside
30 chapters in five parts, 81 drawn figures, hints and solutions
Length
About 37,400 words, about 3 hours straight through; far longer with a pencil
Price
Free during early access
Read free

Sign in and it opens in the reader, on phone, tablet or laptop. Read the opening first.

About the book

Euclid’s Garden starts where Euclid did, with the few words a mathematician agrees not to define, and climbs from there: angles, parallel lines, the 180° triangle, congruence and constructions with compass and straightedge.

The middle of the book is the triangle and the circle, the two objects most olympiad geometry is about. The last part is the olympiad toolbox itself: angle chasing, trigonometry, Ceva and Menelaus, coordinates, vectors and complex numbers, transformations, inversion, and a chapter on how to attack a problem you have never seen.

Each chapter opens with a question to wonder about, and the ideas that matter most are pulled out as “Aha” moments and common traps. The appendices hold a cheat-sheet of theorems, the constructions, and hints and solutions.

What’s inside

The table of contents, as it appears in the book.

Part IFirst Steps

  1. 1Points, Lines, and the Art of Definition
  2. 2Angles and How to Measure Them
  3. 3Parallel Lines and the Transversal
  4. 4Triangles and the 180° Miracle
  5. 5Congruence: When Are Two Things the Same?
  6. 6Compass and Straightedge

Part IIThe Triangle’s Secrets

  1. 7Isosceles, Equilateral, and the Bridge of Asses
  2. 8Similarity and the Power of Ratio
  3. 9Pythagoras, Four Ways
  4. 10The Four Centres
  5. 11Area as a Tool

Part IIIThe Circle

  1. 12Chords, Arcs, and the Circle’s Anatomy
  2. 13The Inscribed Angle Theorem
  3. 14Cyclic Quadrilaterals and Ptolemy
  4. 15Power of a Point and the Radical Axis
  5. 16Tangents and the Tangent–Chord Angle

Part IVThe Toolbox

  1. 17Angle Chasing, the Workhorse
  2. 18Trigonometry in Geometry
  3. 19Ceva, Menelaus, and Concurrency
  4. 20Stewart, Apollonius, and Lengths
  5. 21Coordinates: Geometry by Algebra
  6. 22Vectors and Complex Numbers
  7. 23Transformations

Part VOlympiad Geometry

  1. 24The Configurations You Must Know
  2. 25Concurrency and Collinearity, Top Level
  3. 26Spiral Similarity
  4. 27Inversion: The Great Simplifier
  5. 28The Computational Arts
  6. 29How to Attack an Olympiad Problem
  7. 30The Summit: Olympiad Problems

And afterEpilogue and appendices

  1. Epilogue: The Garden in Bloom
  2. A. Theorem and Formula Cheat-Sheet
  3. B. Constructions
  4. C. Hints and Solutions
  5. D. Further Reading

Read the first pages

The opening of Chapter 1, as it appears in the reader.

Euclid’s GardenChapter 1

Chapter 1Points, Lines, and the Art of Definition

Wonder

Try to define the word “point” to a friend, using only simpler words. Then define those words. Keep going. You will find, surprisingly quickly, that you run out of simpler words entirely. Where does a chain of definitions begin?

Every dictionary is secretly a circle. Look up “line” and it sends you to “straight”; look up “straight” and it sends you back to “line.” For ordinary language this is harmless. For mathematics, where we demand certainty, it is a crisis: if our definitions chase each other in circles, our proofs rest on nothing. The Greeks found the only possible escape, and it is the foundation of everything that follows.

The terms we agree not to define

We choose a tiny handful of words and agree, openly, not to define them. Instead we describe how they behave. These are the primitive (or undefined) terms. In plane geometry there are just three: a point marks a position with no size; a line is straight, thin, and endless in both directions; a plane is a perfectly flat surface extending forever.

Aha!

We do not build mathematics by defining everything. We build it by defining almost everything in terms of a few honest, undefined starting points, and then never sneaking in a hidden assumption again. Knowing where to stop defining is the first act of mathematical maturity.

From these three primitives, every other object is built by definition. For distinct points A and B: the segment AB is A, B, and all points between (finite); the ray AB starts at A and runs through B forever (one endpoint and a direction); the line through them extends forever both ways.

ABlineABrayABsegment
Figure 1.1. The same two points A, B give a line (infinite), a ray (half-infinite), and a segment (finite). The arrowheads mean “continues forever.”

The chapter goes on to the first axiom, distance and the midpoint, and why this much care pays off.

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