Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Euclidean geometry

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.

Euclidean geometry sometimes means geometry in the plane which is also called plane geometry. Plane geometry is the topic of this article.

Euclidean geometry in three dimensions is traditionally called solid geometry. For information on higher dimensions see Euclidean space.

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.

1 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.

The five postulates of the Elements are:

1. Any two points can be joined by a straight line.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circleSee The Circle for the distributed file storage system, and see Ring (diacritic) for the diacritic mark. 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 . can be drawn having the segment as radiusThe word radius ( Latin for "wheel spoke"; plural radii pronounced ray dee-eye has several meanings in English: In classical geometry, a radius of a circle or sphere is any line segment with one endpoint on the circle (i. the circular boundary) and the ot and one endpoint as center.
4. All right angles are congruent.
5. 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 postulateIn geometry, the parallel postulate also called Euclid's fifth postulate since it is the fifth postulate in Euclid's Elements, is a distinctive axiom in what is now called Euclidean geometry. It states: If a line segment intersects two straight lines form, which leads to the same geometry as the statement:

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).

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.

2 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.

Politics	Business	Finance

Topics: Euclidean Geometry Line Axioms Straight Plane Postulate Points...

---

Euclidean geometryEuclidean geometry 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. Euclidean geometry someti Euclidean spaceEuclidean space is the usual n dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer n n dimensional Euclidean space is the set R n (where R is the set of real numbers Euclidean domainIn abstract algebra, a Euclidean domain (also called a Euclidean ring is a type of ring in which the Euclidean algorithm can be used. More precisely, a Euclidean domain is an integral domain D for which can be defined a function v mapping nonzero elements Euclidean algorithmThe Euclidean algorithm (also called Euclid's algorithm is an algorithm to determine the greatest common divisor (gcd) of two integers. It is one of the oldest algorithms known, since it appeared in Euclid's Elements around 300 BC. The algorithm does not Euclidean distanceThe Euclidean distance of two points x x . x and y y . y in Euclidean n space is computed as : It is the "ordinary" distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem Euclidean groupIn mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. It is therefore one of the oldest and most studied groups, at least in the cases of dimension 2 and 3 — implicitly, many of its properties are familiar, if not i Euclidean quantum gravity Euclidean minimum spanning tree