6 Best Geometry Practice Books For Competitive Math That Build Intuition

Your child breezes through algebra, but then they hit a geometry problem in a math competition and just… stare. The formulas they memorized don’t seem to apply, and they can’t see the "trick." This is a common hurdle, where rote learning meets its limit and the need for true geometric intuition becomes clear.

Developing Geometric Intuition for Competitions

Has your child ever looked at a complex diagram and been able to "see" the hidden right triangle or the clever way to draw an auxiliary line? That’s geometric intuition. It’s the ability to visualize relationships, transform shapes in your mind, and sense which path will lead to a solution without testing every single formula.

This skill isn’t magic; it’s built through exposure to well-designed problems that encourage discovery over memorization. For young competitors in middle or early high school, this is a critical developmental step. They are moving from concrete calculations to more abstract reasoning, and geometry is the perfect playground for that growth. The right books act as a guide, showing them how to think like a mathematician.

AoPS Introduction to Geometry for Foundations

You’ve likely heard of Art of Problem Solving (AoPS) if your child is in the competitive math world. Their Introduction to Geometry is the gold standard for a reason, and it’s the perfect place to start for most students aiming for contests like the AMC 8/10 or Mathcounts. It’s the reliable, all-terrain vehicle of geometry texts.

This book’s magic is that it teaches by asking questions. Instead of just presenting a theorem and an example, it guides the student to discover the concepts themselves through carefully sequenced problems. This Socratic method is incredibly effective for building a rock-solid foundation. If you are going to buy just one book to get started, this is it. It aligns perfectly with the problem-solving skills needed for early-to-mid-level competitions.

Kiselev’s Geometry for a Rigorous Approach

Is your child past the basics but struggling to write clean, logical proofs? Do they have good ideas but can’t formalize them? This is where a book like Kiselev’s Geometry (Planimetry I, Stereometry II) comes in. Think of it as the classical instrument that teaches perfect technique.

Originally from Russia, this text is known for its rigorous, axiomatic approach. It builds the entire structure of Euclidean geometry from the ground up, leaving no logical gaps. It is dense and demanding, requiring a level of focus that not every student is ready for. But for the serious high schooler who wants to understand the "why" behind every single rule, Kiselev builds a mental discipline that is unmatched. It’s an investment in deep, logical thinking that pays dividends far beyond geometry.

Geometry Revisited for Deeper Conceptual Links

Sometimes a student knows all the individual theorems but can’t see the beautiful forest for the trees. They can solve for an angle using the inscribed angle theorem, but they don’t appreciate its connection to other circle properties. Geometry Revisited by Coxeter and Greitzer is the book that connects the dots.

This isn’t a textbook you work through from start to finish. It’s more of a guided tour of the greatest hits of geometry, showing how different concepts are elegantly intertwined. It’s ideal for the curious student who has a solid foundation from a book like AoPS but wants to elevate their understanding. It builds intuition by revealing the stunning interconnectedness of geometric ideas, turning isolated facts into a powerful, cohesive toolkit.

Challenging Problems in Geometry for Practice

Your child has the theory down. They understand the theorems and can follow proofs. Now, they need to build speed, pattern recognition, and problem-solving stamina. Challenging Problems in Geometry is the training gym for the aspiring AIME qualifier.

This book is exactly what its title promises: a collection of tough problems, sorted by topic, designed to sharpen skills. It’s not a book for learning new material. It’s for applying what you already know in clever and non-obvious ways. This is a crucial second-stage resource. A student should only dive into this after they have a firm grasp of the fundamentals, otherwise it can be discouraging.

AwesomeMath’s 106 Problems for AMC/AIME Prep

When your child’s goal becomes laser-focused on performing well on the American Mathematics Competitions (AMC) and the American Invitational Mathematics Examination (AIME), you need a specialized tool. 106 Geometry Problems from the AwesomeMath Summer Program is precisely that. It’s like hiring a coach who knows the playbook for a specific opponent.

The problems in this book are curated to mimic the style and difficulty of high-level AMC and AIME questions. The solutions are detailed and often present multiple ways of solving the same problem, which is invaluable for developing flexible thinking. This is a strategic purchase for the student who is consistently scoring well on the AMC 10/12 and is serious about making the jump to the next level.

EGMO by Evan Chen for Olympiad-Level Mastery

For a very small number of students, the goal isn’t just to qualify for an Olympiad, but to excel there. This is a different world of mathematics, focused on proof and deep, original thought. Evan Chen’s Euclidean Geometry in Mathematical Olympiads (EGMO) is the modern bible for these aspirants.

Let’s be clear: this book is for the deeply passionate, self-motivated student who has already mastered the AIME-level material. It covers advanced techniques and ways of thinking that are rarely seen in standard curricula. It teaches a student not just to solve problems, but to think about the very structure of geometry itself. This is the capstone book for a student whose journey has taken them to the highest peaks of high school competitive math.

Integrating These Books into a Study Plan

Seeing this list can feel overwhelming. The key is not to buy them all at once, but to see them as a progression that mirrors your child’s growth. A sensible path looks something like this:

• The Foundation (Grades 7-9): Start and finish AoPS Introduction to Geometry. This is non-negotiable for building the core skills and problem-solving mindset.
• Deepening Understanding (Grades 9-10): Once AoPS is complete, choose a path based on your child’s needs. If they need more rigor and logical structure, go to Kiselev. If they thrive on seeing connections and the "big picture," introduce Geometry Revisited.
• Sharpening for Competition (Grades 10-11): For the student aiming to qualify for the AIME, it’s time for intense practice. Use Challenging Problems in Geometry for broad practice and 106 Geometry Problems for targeted AMC/AIME prep.
• Reaching for the Summit (Grades 11-12): For the established AIME qualifier with USA(J)MO aspirations, EGMO is the final step. It’s a multi-year project, not a book to be rushed.

The most important thing is to match the book to your child’s current stage. Giving an advanced book too early is a recipe for frustration and burnout. The goal is to provide the right challenge at the right time to keep the flame of curiosity alive.

Ultimately, the best book is the one that gets used. Your role isn’t to force a path, but to provide the right tools when your child shows they’re ready for the next step in their journey. Fostering that love of a beautiful problem is the real prize.