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Proof-Based Geometry

H²

Proof-Based Geometry

Euclidean, affine, hyperbolic, and spherical geometries

154Modules

9Units

Being rebuilt

This course is still in our original format. All the content is here and free to use, but it has not yet been through the rebuild that gives our newer courses their interactive animations and cleaner visuals — so you may hit some rough edges. Spot an error? Send us a note and we will fix it.

Recommended preparation

Geometry and a first course in proofs (Intro to Proofs) are the foundation for this rigorous treatment.

Geometry Intro to Proofs

Why Do We Need Axioms?INTRO

Euclid's Elements and Its LegacyINTRO

Undefined Terms and DefinitionsLESSON

Hilbert's Revolution in GeometryINTRO

Axioms of IncidenceLESSON

Axioms of BetweennessLESSON

Axioms of CongruenceLESSON

Proving Theorems from AxiomsEXAMPLE

Axioms of ContinuityLESSON

The Parallel PostulateLESSON

Models of Axiomatic SystemsLESSON

Axiom System ExplorerACTIVITY

Independence ProofsEXAMPLE

Practice: Axiomatic ProofsPRACTICE

Summary: Axiomatic FoundationsSUMMARY

Quiz: Foundations of GeometryQUIZ

The Euclidean PlaneINTRO

Triangle Congruence TheoremsLESSON

Similarity TheoremsLESSON

Parallel Lines and TransversalsLESSON

Circle TheoremsLESSON

Classical Compass and StraightedgeINTRO

Geometric Construction LabACTIVITY

Ptolemy's TheoremEXAMPLE

Theory of AreaLESSON

The Pythagorean Theorem and ProofsLESSON

Real World: Surveying and NavigationREAL WORLD

Ceva's and Menelaus' TheoremsLESSON

The Nine-Point CircleLESSON

Proofs of Triangle Center PropertiesEXAMPLE

Practice: Euclidean ProofsPRACTICE

Summary: Euclidean GeometrySUMMARY

Formula Sheet: Euclidean GeometryFORMULA SHEET

Quiz: Euclidean GeometryQUIZ

What is Affine Geometry?INTRO

Affine vs Euclidean GeometryINTRO

Affine Spaces and SubspacesLESSON

Affine Combinations and CoordinatesLESSON

Affine TransformationsLESSON

Affine Transformation VisualizerACTIVITY

Real World: Computer GraphicsREAL WORLD

Parallel ProjectionLESSON

Ratios and Cross-RatiosLESSON

Barycentric CoordinatesLESSON

Centroid Properties via Affine GeometryEXAMPLE

Affine InvariantsLESSON

Convexity in Affine GeometryLESSON

Proofs Using Affine MethodsEXAMPLE

Practice: Affine GeometryPRACTICE

Summary: Affine GeometrySUMMARY

Formula Sheet: Affine GeometryFORMULA SHEET

Quiz: Affine GeometryQUIZ

Perspective in Art and GeometryINTRO

The Idea of Points at InfinityINTRO

The Projective PlaneLESSON

Homogeneous CoordinatesLESSON

Projective Geometry VisualizerACTIVITY

The Principle of DualityLESSON

Projective TransformationsLESSON

Real World: Computer VisionREAL WORLD

The Cross-RatioLESSON

Conics in Projective GeometryLESSON

Perspective ConstructionsEXAMPLE

Pascal's and Brianchon's TheoremsLESSON

Desargues' TheoremLESSON

Proofs Using Projective MethodsEXAMPLE

Practice: Projective GeometryPRACTICE

Summary: Projective GeometrySUMMARY

Formula Sheet: Projective GeometryFORMULA SHEET

Quiz: Projective GeometryQUIZ

The Problem of Parallel LinesINTRO

Discovery of Non-Euclidean GeometryINTRO

Axioms of Hyperbolic GeometryLESSON

Intuition for Hyperbolic SpaceINTRO

The Poincaré Disk ModelLESSON

Poincaré Disk ExplorerACTIVITY

The Upper Half-Plane ModelLESSON

Lines and GeodesicsLESSON

Hyperbolic DistanceLESSON

Hyperbolic Distance CalculationsEXAMPLE

Hyperbolic TrianglesLESSON

Angle Sum and DefectLESSON

Isometries of Hyperbolic SpaceLESSON

Hyperbolic ConstructionsEXAMPLE

Area in Hyperbolic GeometryLESSON

Tilings of the Hyperbolic PlaneLESSON

Real World: Escher's Circle LimitREAL WORLD

Real World: Special RelativityREAL WORLD

Practice: Hyperbolic GeometryPRACTICE

Summary: Hyperbolic GeometrySUMMARY

Formula Sheet: Hyperbolic GeometryFORMULA SHEET

Quiz: Hyperbolic GeometryQUIZ

Geometry on the Earth's SurfaceINTRO

Great Circles as 'Lines'INTRO

Axioms of Spherical GeometryLESSON

Spherical Lines and DistanceLESSON

Spherical Geometry ExplorerACTIVITY

Spherical TrianglesLESSON

Spherical TrigonometryLESSON

Spherical Law of CosinesLESSON

Spherical Law of SinesLESSON

Great Circle NavigationEXAMPLE

Real World: GPS and NavigationREAL WORLD

Angle Sum and ExcessLESSON

Area of Spherical FiguresLESSON

Lunes and Their PropertiesLESSON

Spherical PolygonsLESSON

Spherical Distance CalculationsEXAMPLE

Real World: Celestial NavigationREAL WORLD

Practice: Spherical GeometryPRACTICE

Summary: Spherical GeometrySUMMARY

Formula Sheet: Spherical GeometryFORMULA SHEET

Quiz: Spherical GeometryQUIZ

The Idea of Elliptic GeometryINTRO

Spherical vs Elliptic GeometryINTRO

The Elliptic PlaneLESSON

Identifying Antipodal PointsLESSON

Lines in Elliptic GeometryLESSON

Elliptic Geometry ExplorerACTIVITY

Distance and Angle MeasureLESSON

Elliptic TrianglesLESSON

Proofs in Elliptic GeometryEXAMPLE

Elliptic TrigonometryLESSON

Isometries of Elliptic SpaceLESSON

Practice: Elliptic GeometryPRACTICE

Summary: Elliptic GeometrySUMMARY

Formula Sheet: Elliptic GeometryFORMULA SHEET

Quiz: Elliptic GeometryQUIZ

Three Worlds of GeometryINTRO

Parallel Lines in Each GeometryLESSON

Triangle Angle Sums ComparedLESSON

Geometry Comparison ToolACTIVITY

Area Formulas ComparedLESSON

Similar Triangles in Each GeometryLESSON

Proofs That Work in All GeometriesEXAMPLE

Trigonometric Formulas ComparedLESSON

Isometry Groups ComparedLESSON

Geometry-Specific PhenomenaEXAMPLE

How Curvature Unifies GeometriesINTRO

Gaussian Curvature PerspectiveLESSON

Real World: Shape of the UniverseREAL WORLD

Practice: Comparing GeometriesPRACTICE

Summary: Comparing GeometriesSUMMARY

Quiz: Geometry ComparisonQUIZ

Complete Course SummarySUMMARY

Euclidean Geometry FormulasFORMULA SHEET

Affine Geometry ReferenceFORMULA SHEET

Projective Geometry ReferenceFORMULA SHEET

Hyperbolic Geometry FormulasFORMULA SHEET

Spherical Geometry FormulasFORMULA SHEET

Elliptic Geometry ReferenceFORMULA SHEET

Proof Techniques in GeometryFORMULA SHEET

Comprehensive Practice ProblemsPRACTICE

Final AssessmentQUIZ