Subject

Geometry

Geometry is at the heart of our teaching philosophy. Spatial reasoning develops the visual intuition that makes every branch of mathematics more accessible. We teach geometry deeply — not as a collection of formulas, but as a system of logical relationships.

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The Geometry pathway

First Primary Junior Intermediate Senior

Ages 16–18 · Years 12–13

Advanced Euclidean and projective geometry

Senior geometry training covers the advanced results seen in national and international Olympiads: power of a point, radical axes, projective transformations and inversions. A deep grounding here is invaluable for BMO and beyond.

What we cover

Power of a point and radical axes
Projective geometry fundamentals
Inversions and Möbius transformations
Advanced trigonometric identities
Synthetic vs analytic proof strategies

Competitions & exams we prepare for in United Kingdom

SMC
BMO
MOG
STEP · MAT · TMUA
IMO

Olympiad curriculum

Durable mathematical ideas form the curriculum; individual puzzles and practice lessons sit beneath them. Topics recur at increasing depth across the pathway.

Advanced Triangle Geometry

Signed Ratios and Directed Lengths
Advanced Ceva and Menelaus Configurations
Symmedians and the Lemoine Point
Isogonal Conjugates
Isotomic Conjugates
Gergonne and Nagel Points
Euler and Nagel Lines
Nine-Point Centre and Feuerbach's Theorem
Morley's Trisector Theorem
Triangle-Centre Configurations

Advanced Circle Geometry

Radical-Axis and Radical-Centre Configurations
Coaxal Systems of Circles
Miquel Points
Complete Quadrilaterals and Miquel Circles
Simson and Steiner Lines
Reim's Theorem
Monge's Theorem for Three Circles
Poncelet Porism: Introduction
Tangent-Circle and Incircle Networks
Advanced Power-of-a-Point Configurations

Transformational Geometry

InversionOpens on Problems.cc
Choosing an Inversion Centre and Radius
Inversion with Orthogonal Circles
Spiral Similarity Centres
Composition of Rotations and Homotheties
Möbius Transformations: Introduction
Transformations between Lines and Circles

Projective Geometry

Cross-Ratios and Harmonic Division
Complete Quadrilaterals
Pole and Polar with Respect to a Circle
La Hire's Theorem
Pascal's Theorem
Brianchon's Theorem
Projective Transformations and Points at Infinity

Analytic Olympiad Methods

Barycentric Coordinates
Trilinear Coordinates
Complex Numbers in Geometry
Unit-Circle Complex Methods
Vector and Dot-Product Methods
Advanced Trigonometric Geometry
Cartesian and Polar Coordinate Methods

Inequalities, Extremal & Convex Geometry

Erdős–Mordell Inequality
Ptolemy's Inequality
Geometric Inequalities by Area and Trigonometry
Isoperimetric Problems for Polygons
Fermat–Torricelli and Geometric Median Problems
Convex Sets and Convex Hulls
Olympiad Problems in Covering and Packing

Geometry

This course encompasses a range of Geometry topics such as coordinate and spatial geometry, introductory trigonometry, angles, parallel lines, congruent and similar triangles, polygons, circles, the Pythagorean Theorem, and more. Emphasis will be placed on reinforcing Algebra skills and enhancing critical thinking through problem-solving in both mathematical and real-world contexts.

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Articles

All geometry posts →

Geometry3 min read

Diving Deep into Geometry: Exact Science's Camp Goes Beyond the Classroom

Why serious geometry still needs more proof, construction, and precision than most students ever get in school, and why that matters for mathematical maturity.

George Ionitsa·19 February 2024

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Geometry

Shapes, symmetry and visual puzzles

Shapes, symmetry and spatial intuition

Euclidean reasoning and proof

Circle theorems and classical constructions

Advanced Euclidean and projective geometry

Geometry

Diving Deep into Geometry: Exact Science's Camp Goes Beyond the Classroom

Start your Geometry journey