Curriculum
Topics we teach — and have taught
These pages summarise how Exact Science structures olympiad-style mathematics and programming: the core ideas, worked theory, and problem sets we have used in classes over the years. Many topics now also have interactive practice on Problems.cc.
Browse interactive lessonsGeometry
Proofs, constructions, and spatial reasoning that go beyond standard school exams.
All levels
Axioms and Postulates of Euclid
Minimum and Maximum Problems in Geometry
Minimum and Maximum Problems in Geometry
Challenging Triangle Congruence
Most students are already familiar with the three standard triangle congruence rules taught in school: SSS (Side-Side-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle). These form the foundation of geometry and are commonly seen in classroom exercises and exams, including the GCSE Mathematics syllabus.But what happens when we encounter problems that don’t follow these standard patterns directly? Can we still prove triangle congruence using medians, altitudes, angle bisectors, or partial side information?In this section, we explore a selection of challenging triangle congruence problems that require deeper reasoning and clever constructions. These problems go beyond the textbook basics and are perfect for students preparing for Olympiad-style questions, GCSE extensions, or anyone wanting to sharpen their proof skills.
Circumcircle of a Triangle
Practice problems on circumcircles of triangles: centers, angles, chords, and construction. Ideal for Olympiad geometry and advanced learners.
Inscribed Quadrilateral
Learn the theory and solve challenging problems about inscribed quadrilaterals. Perfect for math Olympiad prep, this page covers trapezoids, kites, and key geometric proofs.
Morley's Triangle
Opens on Problems.ccTangent
Explore key theorems about tangents to circles and solve problems involving radii, angles, and geometric constructions with tangents.