08-20-2026, 02:39 PM
Book:Introduction to Geometry
Author: Richard Rusczyk
Publisher: AoPS Incorporated (Art of Problem Solving)
Original publication: 2006
Edition: 2nd edition, 2009
Length: 557-page textbook + 226-page solutions manual
ISBN: 978-1-934124-08-6
Level: Roughly grades 7–10 / strong middle-school to early high-school mathematics
Summary
Introduction to Geometry is not a conventional school geometry textbook. Richard Rusczyk organizes the material around problem solving and mathematical discovery rather than presenting a theorem followed immediately by routine exercises. Sections typically begin with problems that students are encouraged to attack before reading the explanation. The subsequent discussion develops the relevant ideas through solutions, observations, proofs, and general techniques. This makes the book particularly suitable for students who want to understand why geometric results work and how to recognize when to use them, rather than simply memorize formulas. The book contains more than 900 problems, with complete solutions available in the accompanying solutions manual.
The mathematical coverage is broad enough to serve as a complete introductory geometry course. It develops angles and parallel lines, formal proof, triangle congruence and similarity, right triangles, areas, perpendicular bisectors, altitudes, angle bisectors, medians and concurrency, quadrilaterals, polygons, geometric inequalities, circles, tangents, power of a point, three-dimensional geometry, geometric transformations, coordinate/analytic geometry, and introductory trigonometry. The associated AoPS curriculum goes as far as the Laws of Sines and Cosines and trigonometric identities, while also emphasizing general geometry problem-solving strategies.
A major strength is that elementary ideas are repeatedly pushed into non-routine and competition-style problems. The intended audience includes students preparing for MATHCOUNTS and the AMC 10/12, although competition participation is not necessary to benefit from it. AoPS recommends that students already be comfortable with basic algebra, particularly solving linear equations. Consequently, the book sits somewhere between a rigorous school geometry course and an introduction to mathematical-competition geometry: substantially more challenging than most ordinary secondary-school textbooks, but not yet an olympiad-geometry text.
Main topics
The progression is approximately:
Foundations & triangles
Angles → parallel lines → proofs → congruent triangles → isosceles/equilateral triangles → similarity → right triangles.
Classical Euclidean geometry
Area → perpendicular bisectors → altitudes → angle bisectors → medians → concurrency → Ceva's theorem → quadrilaterals → polygons → geometric inequalities.
Circles
Arcs and sectors → angles in circles → tangent lines → secants → chords → power of a point.
Beyond elementary plane geometry
3-D geometry → transformations → analytic/coordinate geometry → introductory trigonometry → Laws of Sines and Cosines → trigonometric identities → general problem-solving strategies.
What makes the book distinctive
The central philosophy is essentially:
problem → experiment → discover an idea → prove it → use it on harder problems.
That is quite different from the more common:
definition → theorem → worked example → repetitive exercises.
Rusczyk's approach encourages students to develop habits that are extremely useful in higher mathematics: drawing auxiliary lines, looking for similar triangles, translating geometric relationships into algebra, identifying invariants, working backwards from the desired result, and combining several elementary facts rather than searching for a single memorized theorem. AoPS itself describes the course as helping students transition toward a more mature view of mathematics and formal proof.
Who should read it?
I would particularly recommend it for:
Key takeaways
AoPS — Introduction to Geometry
Author: Richard Rusczyk
Publisher: AoPS Incorporated (Art of Problem Solving)
Original publication: 2006
Edition: 2nd edition, 2009
Length: 557-page textbook + 226-page solutions manual
ISBN: 978-1-934124-08-6
Level: Roughly grades 7–10 / strong middle-school to early high-school mathematics
Summary
Introduction to Geometry is not a conventional school geometry textbook. Richard Rusczyk organizes the material around problem solving and mathematical discovery rather than presenting a theorem followed immediately by routine exercises. Sections typically begin with problems that students are encouraged to attack before reading the explanation. The subsequent discussion develops the relevant ideas through solutions, observations, proofs, and general techniques. This makes the book particularly suitable for students who want to understand why geometric results work and how to recognize when to use them, rather than simply memorize formulas. The book contains more than 900 problems, with complete solutions available in the accompanying solutions manual.
The mathematical coverage is broad enough to serve as a complete introductory geometry course. It develops angles and parallel lines, formal proof, triangle congruence and similarity, right triangles, areas, perpendicular bisectors, altitudes, angle bisectors, medians and concurrency, quadrilaterals, polygons, geometric inequalities, circles, tangents, power of a point, three-dimensional geometry, geometric transformations, coordinate/analytic geometry, and introductory trigonometry. The associated AoPS curriculum goes as far as the Laws of Sines and Cosines and trigonometric identities, while also emphasizing general geometry problem-solving strategies.
A major strength is that elementary ideas are repeatedly pushed into non-routine and competition-style problems. The intended audience includes students preparing for MATHCOUNTS and the AMC 10/12, although competition participation is not necessary to benefit from it. AoPS recommends that students already be comfortable with basic algebra, particularly solving linear equations. Consequently, the book sits somewhere between a rigorous school geometry course and an introduction to mathematical-competition geometry: substantially more challenging than most ordinary secondary-school textbooks, but not yet an olympiad-geometry text.
Main topics
The progression is approximately:
Foundations & triangles
Angles → parallel lines → proofs → congruent triangles → isosceles/equilateral triangles → similarity → right triangles.
Classical Euclidean geometry
Area → perpendicular bisectors → altitudes → angle bisectors → medians → concurrency → Ceva's theorem → quadrilaterals → polygons → geometric inequalities.
Circles
Arcs and sectors → angles in circles → tangent lines → secants → chords → power of a point.
Beyond elementary plane geometry
3-D geometry → transformations → analytic/coordinate geometry → introductory trigonometry → Laws of Sines and Cosines → trigonometric identities → general problem-solving strategies.
What makes the book distinctive
The central philosophy is essentially:
problem → experiment → discover an idea → prove it → use it on harder problems.
That is quite different from the more common:
definition → theorem → worked example → repetitive exercises.
Rusczyk's approach encourages students to develop habits that are extremely useful in higher mathematics: drawing auxiliary lines, looking for similar triangles, translating geometric relationships into algebra, identifying invariants, working backwards from the desired result, and combining several elementary facts rather than searching for a single memorized theorem. AoPS itself describes the course as helping students transition toward a more mature view of mathematics and formal proof.
Who should read it?
I would particularly recommend it for:
- strong students aged roughly 12–16;
- students who find ordinary school geometry too mechanical;
- students preparing for AMC 8/10/12 or MATHCOUNTS;
- teachers looking for challenging enrichment problems;
- adults wanting to relearn Euclidean geometry from a problem-solving perspective.
Key takeaways
- Geometry is taught as problem solving, not memorization. The student is expected to discover many ideas by attempting problems first.
- The coverage is unusually comprehensive: classical Euclidean geometry, proofs, circles, transformations, analytic geometry, 3-D geometry and introductory trigonometry all appear in one course.
- The difficulty rises well beyond ordinary textbook exercises. Its 900+ problems make it particularly valuable for mathematically ambitious students and competition preparation.
- It is an excellent bridge toward olympiad mathematics, although students aiming specifically at high-level olympiad geometry will eventually need a more advanced text.
AoPS — Introduction to Geometry
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