Euclidean geometry

plane geometryEuclideanEuclidean plane geometryEuclid's postulatesplanarplaneclassical geometryEuclid's axiomsEuclidean geometricEuclidean plane
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.wikipedia
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Euclid's Elements

ElementsEuclid's ''ElementsEuclid
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines.

Euclid

Euclid of AlexandriaEuklidGreek Mathematician
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
In the Elements, Euclid deduced the theorems of what is now called Euclidean geometry from a small set of axioms.

Non-Euclidean geometry

non-Euclideannon-Euclidean geometriesalternative geometries
Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century.
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

Parallel postulate

fifth postulateEuclid's fifth postulateEuclid's parallel postulate
Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry.

Pythagorean theorem

Pythagoras' theoremPythagorasPythagoras's theorem
(Book 1 proposition 17) and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle."
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

Foundations of geometry

foundationsaxiomatic geometryPeano's axiom system
Modern treatments use more extensive and complete sets of axioms.
There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries.

Formal system

logical systemdeductive systemsystem of logic
Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system.
Often the formal system will be the basis for or even identified with a larger theory or field (e.g. Euclidean geometry) consistent with the usage in modern mathematics such as model theory.

Circle

circularcircles360 degrees
This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted.

Point (geometry)

pointpointslocation
More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built, meaning that a point cannot be defined in terms of previously defined objects.

Mathematical proof

proofproofsprove
The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof.
Alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate.

Geometry

geometricgeometricalgeometries
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
In Euclidean geometry, angles are used to study polygons and triangles, as well as forming an object of study in their own right.

General relativity

general theory of relativitygeneral relativity theoryrelativity
An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).
Space, in this construction, still has the ordinary Euclidean geometry.

Rectangle

rectangularoblongrectangles
For example, a rectangle with a width of 3 and a length of 4 has an area that represents the product, 12.
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles.

Similarity (geometry)

similarsimilaritysimilar triangles
Figures that would be congruent except for their differing sizes are referred to as similar.
In Euclidean geometry, two objects are similar if they both have the same shape, or one has the same shape as the mirror image of the other.

Playfair's axiom

Euclid's AxiomPlayfair's formPlayfair's postulate
For example, Playfair's axiom states:
It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry and was named after the Scottish mathematician John Playfair.

Angle

acute angleobtuse angleoblique
Euclidean geometry has two fundamental types of measurements: angle and distance.
In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

Line (geometry)

linestraight linelines
Modern school textbooks often define separate figures called lines (infinite), rays (semi-infinite), and line segments (of finite length).
Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as non-Euclidean, projective and affine geometry).

Synthetic geometry

syntheticpure geometrysynthetic geometer
Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects.
The close axiomatic study of Euclidean geometry led to the construction of the Lambert quadrilateral and the Saccheri quadrilateral.

Axiomatic system

axiomatizationaxiomatic methodaxiom system
The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of formal proof.
Euclid of Alexandria authored the earliest extant axiomatic presentation of Euclidean geometry and number theory.

Euclidean distance

Euclidean metricEuclideandistance
Euclidean geometry has two fundamental types of measurements: angle and distance.
In the context of Euclidean geometry, a metric is established in one dimension by fixing two points on a line, and choosing one to be the origin.

Straightedge and compass construction

compass and straightedgecompass and straightedge constructionscompass-and-straightedge construction
Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge.
(This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with collapsing compass; see compass equivalence theorem.) More formally, the only permissible constructions are those granted by Euclid's first three postulates.

Axiom

axiomspostulateaxiomatic
Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these.
In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., parallel postulate in Euclidean geometry).

Area

surface areaArea (geometry)area formula
Measurements of area and volume are derived from distances.
The mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle.

Affine geometry

affineaffine ''d''-spaceaffine geometries
Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle (whence right triangles become meaningless) and of equality of length of line segments in general (whence circles become meaningless) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments (so line segments continue to have a midpoint).
In mathematics, affine geometry is what remains of Euclidean geometry when not using (mathematicians often say "when forgetting" ) the metric notions of distance and angle.