Plane Euclidean Geometry: Theory and Problems [Gardiner]

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Book name:Plane Euclidean Geometry: Theory and Problems
Authors: A. D. Gardiner (Anthony Gardiner) and C. J. Bradley (Christopher John Bradley)
Publication date: 2012, revised and improved 2nd edition
Publisher: United Kingdom Mathematics Trust (UKMT)
ISBN: 978-1-906001-18-6 / 1-906001-18-9
Length: approximately 210–213 pages
Area: Euclidean geometry / mathematical olympiad problem solving 

Plane Euclidean Geometry: Theory and Problems is a problem-oriented introduction to classical Euclidean geometry aimed particularly at strong secondary-school students and mathematics competition participants. Rather than presenting geometry merely as a collection of formulas and standard theorems, Gardiner and Bradley try to develop geometrical reasoning and mathematical thinking. The exposition broadly follows the historical development associated with Euclid and gradually moves from fundamental geometric ideas toward techniques useful in substantially harder problems. UKMT describes it as intended to make Euclidean geometry accessible to a wider group of younger mathematicians, while still providing material appropriate for challenging competition problems.

The book covers the classical geometry of triangles, circles and configurations of lines, but extends considerably beyond ordinary school geometry. Important topics include the Pythagorean theorem, trigonometry, circle theorems, Ceva's theorem, Menelaus' theorem, geometrical inequalities, and coordinate geometry. These results are developed not simply as isolated facts but as tools for solving problems. The emphasis is on learning how to recognize useful configurations, introduce auxiliary lines, exploit ratios and cyclic structures, and construct rigorous proofs. UKMT material describes the book as containing hundreds of problems, many accompanied by hints or solutions, which makes it suitable for systematic self-study. 

The level is especially appropriate for able students roughly 16+ who already know elementary school geometry and want to progress toward mathematical-olympiad geometry. Christopher Bradley had extensive experience training students for the International Mathematical Olympiad, particularly in geometry, while Tony Gardiner was heavily involved in mathematical enrichment and competition mathematics. The British Mathematical Olympiad specifically recommends the book for BMO preparation, highlighting chapters 3–7 as particularly useful. 

Key points

• Strong bridge between school geometry and Olympiad geometry.
  
• Develops proof and problem-solving skills rather than relying on memorized formulas.
  
• Covers central competition tools such as Ceva, Menelaus, circle geometry and geometric inequalities.
  
• Best suited to students who already know elementary geometry and want substantially harder problems.
  
• Particularly relevant for UKMT/BMO-style competitions, but its techniques apply much more broadly to geometry contests. 
  

Assessment: This is a particularly good choice for someone wanting a relatively systematic introduction to Olympiad-level Euclidean geometry. It is less encyclopedic than some advanced geometry texts, but that is an advantage for competition preparation: the focus stays on useful theorems, geometric insight, and solving progressively harder problems.

Official UKMT book page