Euclidean geometry

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

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.

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 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.