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In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote quotations related to: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...
Euclidean geometry sometimes means geometry in the plane which is also called plane geometry. Plane geometry is the topic of this article. Euclidean geometry is also based off of the Point-Line-Plane postulate. Euclidean geometry in three dimensions is traditionally called solid geometry. For information on higher dimensions see Euclidean space. In mathematics, a plane is the fundamental two-dimensional object. ...
The point-line-plane postulate in geometry is a collective of three assumptions (axioms) that are the basis for Euclidean geometry in three dimensions (solid geometry) or more. ...
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. ...
In mathematics and astronomy, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...
Plane geometry is the kind of geometry usually taught in high school. Euclidean geometry is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ...
Japanese high school students in uniform High school, or Secondary school, is the last segment of compulsory education in Australia, Canada, Hong Kong, Japan, Malaysia, South Korea, Singapore, Taiwan (Republic of China) (only junior high school) and the United States. ...
Euclid of Alexandria (Greek: ) (circa 365–275 BC) was a Greek mathematician, now known as the father of geometry. He was probably alive during the reign of Ptolemy I, (306-233 B.C.E). ...
Euclids Elements (Greek Στοιχεία) is a mathematical and geometric treatise, consisting of 13 books, written by the Greek mathematician Euclid around 300 BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ...
Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Thales (circa 624-547 BC) dealing with spatial relationships. ...
Axiomatic approach
The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...
A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ...
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
The five postulates of the Elements are: For the algebra software named Axiom, see Axiom computer algebra system. ...
- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The fifth postulate is called the parallel postulate, which leads to the same geometry as the statement: The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution...
A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...
In mathematics, a line segment is a part of a line that is bounded by two end points. ...
In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, called the centre. ...
In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. ...
This article is about angles in geometry. ...
See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ...
In geometry, an internal angle is an angle that 2 sides of a polygon form by touching. ...
In geometry, the parallel postulate, also called Euclids fifth postulate since it is the fifth postulate in Euclids Elements, is a distinctive axiom in what is now called Euclidean geometry. ...
- Through a point not on a given straight line, one and only one line can be drawn that never meets the given line.
The parallel postulate seems less obvious than the others and many geometers tried in vain to prove it from them. In the 19th century it was shown that this could not be done, by constructing hyperbolic geometry where the parallel postulate is false, while the other axioms hold. (If one simply drops the parallel postulate from the list of axioms then you get more general geometry called absolute geometry). Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ...
The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
Absolute geometry is a geometry that doesnt assume the parallel postulate. ...
Another thing that was observed was that Euclid's five axioms are actually somewhat incomplete. For instance, one of his theorems is that any line segment is part of a triangle; he constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as third vertex. His axioms, however, do not guarantee that the circles actually intersect. Many revised systems of axioms were constructed, the most standard ones are Hilbert's axioms and Birkhoff's axioms. David Hilberts axioms are a set of 20 assumptions (originally 21) designed to form the foundation for a modern treatment of Euclidean geometry. ...
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoffs axioms. ...
Euclid also had five "common notions" which can also be taken to be axioms, though he later used other properties of magnitudes. In science, magnitude refers to the numerical size of something: see orders of magnitude. ...
- Things which equal the same thing also equal one another.
- If equals are added to equals, then the wholes are equal.
- If equals are subtracted from equals, then the remainders are equal.
- Things which coincide with one another equal one another.
- The whole is greater than the part.
Modern introduction to Euclidean geometry Today, Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. If one introduces geometry this way, one can then prove the Euclidean (or any other) axioms as theorems in this particular model. This does not have the beauty of the axiomatic approach, but it is extremely concise. In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry, is the study of geometry using the principles of algebra. ...
The construction First let us define the set of points as set of pairs of real numbers (x,y). Then given two points P = (x,y) and Q = (z,t) one can define distances using the following formula: The text or formatting below is generated by a template which has been proposed for deletion. ...
- .
This is known as the Euclidean metric. All other notions as a straight line, angle, circle can be defined in terms of points as pairs of real numbers and the distances between them. For example straight line through P and Q can be defined as a set of points A such that the triangle APQ is degenerate, i.e. In mathematics, a metric space is a set (or space) where a distance between points is defined. ...
- .
Classical theorems Cevas Theorem (pronounced Cheva) is a very popular theorem in elementary geometry. ...
In geometry, Herons formula (also called Heros formula) states that the area of a triangle whose sides have lengths a, b and c is where s is the triangles semiperimeter: (see also square root). ...
In geometry, the nine point circle is a circle that can be constructed for any given triangle. ...
There are thousands of proofs of the Pythagorean theorem. ...
Niccolo Fontana Tartaglia. ...
See also Interactive geometry software (IGS) allows you to create and then manipulate geometric constructions, primary in plane geometry. ...
The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes both hyperbolic and elliptic geometry, which are contrasted with Euclidean geometry. ...
External links - Geometry Step by Step from the Land of the Incas (http://agutie.homestead.com) by Antonio Gutierrez.
- Euclid's elements (http://aleph0.clarku.edu/~djoyce/java/elements/toc.html)
- Geometry (http://www.cut-the-knot.org/geometry.shtml)
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