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Euclidean geometry

For a topic other than geometry whose name includes the word "Euclidean", see Euclidean algorithm.

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 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.

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.

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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 circle can be drawn having the segment as radius 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 postulate, 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.

Euclid also had five "common notions" which can also be taken to be axioms, though he later used other properties of magnitudes.

1. Things which equal the same thing also equal one another.
2. If equals are added to equals, then the wholes are equal.
3. If equals are subtracted from equals, then the remainders are equal.
4. Things which coincide with one another equal one another.
5. 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.

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:

$|PQ|=\sqrt{(x-z)^2+(y-t)^2}$.

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.

$|PQ| =|PA|+|AQ| \mbox{ or } |PQ| =\pm(|PA|-|AQ|)$.