- 01Purpose of the edition and modern geometry
- 02Assistance from Professor Townsend
- 03Questions for examination and exercises
- 04Revisions in the second edition
- 05Careful proofreading and intended use
Elements
by Euclid · c. 300 BCE · 30 chapters · 24/7 tutor
Axiomatic reasoning and foundational geometry.
30Chapters
109Lessons
278Topics
Certificate of completion included
This course was produced by Cavua from a public-domain text.
Skills this course leaves you with.
- Apply Euclid's axioms and postulates to build geometric arguments
- Prove basic theorems about angles, triangles, and parallel lines
- Solve classical construction and theorem problems from Book I and Book II
- Use properties of rectangles and parallelograms in geometric proofs
- Analyze circle definitions and derive results from propositions on circles
Curriculum
The full table of contents.
Every chapter, every lesson, every topic — laid out before you enroll. Click any chapter to expand it.
- Chapters
- 30
- Lessons
- 109
- Topics
- 278
- 01Contents
- 01Geometry as figured space and plane geometry
- 02Point, line, plane, and plane figures
- 03Angles, concurrent lines, and triangles
- 01Polygon definitions and classifications
- 02Circle, radius, and diameter
- 03Euclid's postulates
- 04Basic axioms of equality
- 01Axioms and superposition
- 02Theorems, problems, and converse theorems
- 01Prop. I.—Problem
- 01Prop. II.—Problem
- 01Triangle congruence by superposition
- 02Isosceles triangle base angles
- 01Corollary: equilateral triangles are equiangular
- 02Axis of symmetry
- 03Exercises
- 01Prop. VI — Equal angles in a triangle imply equal opposite sides
- 02Prop. VII — Unequal conterminous sides on the same base
- 01Triangle congruence by side-side-base
- 02Bisecting a rectilineal angle
- 01Perpendicular from a point to a line
- 02Adjacent angles on a straight line
- 03Continuation of a straight line
- 04Opposite angles formed by intersecting lines
- 05Exterior angle and triangle angle inequalities
- 01Theorem: greater angle opposite greater side
- 02Triangle side-sum theorem
- 01Triangle inequality for two lines from a point
- 02Construction of a triangle from three given sides
- 01Triangle inequality and circle intersection conditions
- 02Constructing an angle equal to a given angle at a point on a line
- 03Greater side opposite greater angle in triangles
- 04Greater base implies greater contained angle
- 01Congruence of triangles with two equal angles and a side
- 01Definitions of parallel lines and related quadrilaterals
- 02Alternate, corresponding, and interior angles
- 03Criteria for parallel lines from angle equality
- 04Angle properties of parallel lines
- 05Exercises on parallel lines
- 06Transitive parallelism and drawing a parallel through a point
- 01Construction of a parallel line by equal angles
- 02External angle theorem and angle-sum corollaries
- 01Exercises on perpendiculars, angle bisectors, and triangle constructions
- 02Equal and parallel joining lines
- 03Parallelogram properties and converses
- 01Corollaries on parallelograms and lozenges
- 02Exercises on parallelograms and quadrilaterals
- 03Parallelograms on the same base and between the same parallels
- 01Equal parallelograms on equal bases
- 02Triangles on the same base between the same parallels
- 03Two triangles on equal bases between the same parallels
- 01Intercepts and midpoint triangle area results
- 02Equal triangles between the same parallels
- 03Altitude of a triangle
- 04Exercises on triangle and quadrilateral properties
- 05Parallelogram and triangle area relation
- 01Equal areas with equal altitudes
- 02Constructing a parallelogram equal to a triangle
- 03Parallelogram complements on a diagonal
- 01Collinearity of the middle points of three diagonals
- 02Apply a parallelogram equal to a given triangle
- 01PROP. XLV.—PROBLEM
- 01Constructing a square on a given line
- 02Pythagorean theorem for a right triangle
- 01Exercises on right-triangle and square relations
- 02Converse of the Pythagorean theorem
- 03Questions for Examination on Book I
- 01Book I exercises on triangles, parallels, and loci
- 01Preliminary explanations of square measure
- 01Rectangle area as product of sides
- 02Definitions of segments, rectangles, and gnomons
- 03Prop. I: rectangle of a whole line and parts
- 01Distributive law for rectangles
- 02Triangle area as half a rectangle
- 03Square on a divided line
- 01Square on a divided line
- 01Corollaries on squares and line segments
- 02Proposition V: rectangle of unequal parts and square on the middle segment
- 01PROP. VI.—THEOREM
- 01Proposition VII: sum of squares and rectangle
- 01Square on the sum and difference of two lines
- 02Prop. VIII: line divided into two parts
- 03Direct sequence and exercises
- 04Prop. IX: bisected line divided unequally
- 01External division of a bisected line
- 01Exercises on sums of squares and loci
- 02Division of a line in extreme and mean ratio
- 01Incommensurability of extreme and mean ratio segments
- 02Exercises on extreme and mean ratio
- 03Obtuse triangle theorem on the square of the opposite side
- 01Theorem on an acute triangle and the perpendicular intercept
- 01Constructing a square equal to a given rectilineal figure
- 01Definitions of circle elements and related figures
- 01Concentric circles, concyclic points, cyclic quadrilaterals, and directed angles
- 02Finding the centre of a circle
- 03A chord lies inside the circle and outside its extension
- 04A diameter perpendicular to a chord bisects it
- 05Two chords cannot bisect each other
- 01Chords that bisect each other and non-concentric circles
- 02Lines from an interior point to a circle
- 03Lines from an exterior point to a circle
- 01Equal lines from a point to a circle
- 02Centers of circles and contact theorems
- 03Equal chords in equal circles
- 01Greatest chord and chord lengths by distance from the centre
- 01Perpendicularity of the tangent to the radius
- 02Construction of a tangent from an external point
- 01Perpendicular from the centre to a tangent
- 02Converse theorem for tangents through the point of contact
- 01Angle at the centre and angle in the circumference
- 02Angles in the same segment
- 01Opposite angles of a cyclic quadrilateral
- 01Similar segments on the same chord
- 02Similar segments on equal chords
- 03Constructing a circle from an arc
- 04Memory scheme for Propositions XXVI–XXIX
- 01Equal angles and equal arcs in equal circles
- 01Equal arcs subtend equal chords
- 02Bisecting a given arc
- 03Angles in a semicircle and segments
- 01Tangent and chord angles in alternate segments
- 01Angle between a tangent and a chord
- 02Constructing a circular segment with a given angle
- 03Cutting off a segment from a given circle
- 01Intersecting chords and segment rectangles
- 02Concyclic extremities from equal segment products
- 03Equiangular triangles and noncorresponding sides
- 04Exercises on chord products and secants
- 01Secant-segment rectangle constant for a fixed point
- 02Converse tangent criterion
- 01Tangent from a point to a circle
- 02Exercises and examination questions on Book III
- 01Definitions of inscription, circumscription, and regular figures
- 02Proposition I: chord equal to a given line in a circle
- 03Proposition II: inscribing a triangle equiangular to a given triangle
- 01Constructing a triangle equiangular to a given triangle
- 02Inscribing a circle in a given triangle
- 01Circumcircle of a triangle
- 01Inscribing a square in a given circle
- 02Describing a square about a given circle
- 03Inscribing a circle in a given square
- 01Circle described about a given square
- 02Isosceles triangle with base angles double the vertical angle
- 03Inscribing a regular pentagon in a given circle
- 01Regular pentagon circumscribed about a circle
- 02Inscribing a circle in a regular pentagon
- 01Circumscribed circle about a regular pentagon
- 02Inscribing a regular hexagon in a given circle
- 01Inscribing a regular pentagon and equilateral triangle
- 02Regular polygon of fifteen sides
- 03Questions for Examination on Book IV
- 01Euclid’s definitions of ratio and proportion
- 02Continued proportion and compound ratio
- 01Compound, duplicate, and triplicate ratios
- 02Homologous terms and derived proportion propositions
- 01Invertendo: equal ratios and reciprocals
- 02Multiple and submultiple ratios
- 03Equal magnitudes and ratios to a third magnitude
- 04Comparing ratios of unequal magnitudes
- 05Equal ratios imply equal magnitudes
- 06Greater ratio implies greater magnitude
- 01Ratios equal to the same ratio
- 02Sum of equal ratios
- 03Comparison of equal ratios
- 04Antecedent and consequent comparison
- 01Equimultiples and proportional transformations
- 01Proposition XIX: remainder ratios
- 02Proposition XX: excess above the second
- 03Proposition XXI: comparison of two triples
- 01Proposition XXI: transverse order ratios and comparison of extremes
- 02Proposition XXII: ex aequali and continued proportion
- 03Proposition XXIII: ex aequo perturbato
- 01Ratios compounded of equal ratios
- 02Addition of magnitudes in proportion
- 03Sum of extremes and means in proportionals
- 04Questions and exercises on Book V
- 01Definitions of similar figures and proportion
- 02Triangles and parallelograms with the same altitude
- 01Proportional segments in a triangle
- 02Angle bisector theorem
- 03Exercises on proportional division and angle bisectors
- 01Harmonic division properties
- 02Proportional sides of equiangular triangles
- 03Parallel lines meeting at infinity
- 04Exercises
- 01Triangles with proportional sides are equiangular
- 02One equal angle and proportional sides about it
- 03One equal angle, proportional sides, and same-species remaining angles
- 01Right triangle altitude and similar triangles
- 02Cutting off submultiples and dividing lines similarly
- 03Third and fourth proportionals
- 04Mean proportional constructions and applications
- 05Reciprocal proportionality of parallelograms and triangles
- 06Rectangles and squares in proportion
- 07Exercises on chord and rectangle relations
- 01Construction of similar polygons
- 02Directly and inversely similar figures
- 03Corollaries on similar polygons
- 04Similar triangles and duplicate ratio of areas
- 01Similar polygons divided into similar triangles
- 01Homologous points in similar figures
- 01Centre of similitude and homothetic figures
- 01Similar rectilineal figures similar to the same figure
- 02Proportional figures on proportional lines
- 01PROP. XXIII.—T H E O R E M .
- 01Similar parallelograms with a common angle
- 02Maximum parallelogram in a triangle
- 03Inscribing a parallelogram equal to a given figure
- 01Construction of an escribed parallelogram equal to a given figure
- 01Division of a line in extreme and mean ratio
- 02Similar figures on the sides of a right triangle
- 03Parallel homologous sides in joined triangles
- 01Angles, arcs, and sectors in equal circles
- 02Questions for Examination on Book VI
- 03Mean position and least-squares properties
- 04Construction and locus problems
- 05Transversal and collinearity theorems
- 06Perpendicular bisectors and triangle rectangles
- 07Cyclic quadrilateral and tangent-ratio loci
- 08Pascal’s theorem
- 01Two sides and given area of a triangle
- 02Cyclic quadrilateral area and related triangle identities
- 03Parallel-line and circle locus problems
- 04Regular polygons, equilateral-triangle constructions, and triangle metric relations
- 01Variable circle and tangent intercepts
- 01Locus of a point from perpendicular feet on a polygon
- 01Definitions of coplanar lines, dihedral angles, and solid angles
- 02Propositions on planes and intersecting lines
- 03Perpendiculars to coplanar lines and the normal
- 04Three concurrent lines with a common perpendicular
- 01Normal lines to a plane
- 02Projection of a line on a plane
- 03Parallel lines and a transversal are coplanar
- 04A parallel line to a normal is also normal
- 01Parallel lines and angle theorems
- 02Normals to planes and uniqueness
- 03Parallel planes from a common normal
- 04Parallel planes cut by a third plane
- 05Proportional segments in parallel planes
- 01Plane perpendicular to a normal line
- 02Common section of two perpendicular planes
- 03Sum of two face angles of a trihedral angle
- 04Sum of the face angles of a solid angle
- 01Definitions of polyhedra and solids of revolution
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01How many chapters are in the Elements course?
The Elements course has 30 chapters, each with its own lessons and topics.
02How many lessons are in the Elements course?
There are 109 lessons across the course, and you can ask your 24/7 tutor about any of them.
03Can I see what the Elements course covers before I start?
Yes. The full curriculum is listed on this page, beginning with "PREFACE.".
04Do I get a certificate for the Elements course?
Yes. Finishing the course earns a personal Cavua certificate of completion.
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