# Geometry - Proof, Constructions, Points, Lines, And Planes, Angles, Parallel Lines And Planes

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ideas formal system

Geometry, the study of points, lines, and other figures in **space**, is a very old branch of **mathematics**. Its ideas were undoubtedly used, intuitively if not formally, from earliest times. Walking along a straight line toward a particular destination is the shortest way to get there; lining an arrow up with the target is the way to hit it; sitting in a **circle** around a fire is the most equitable way to share the warmth. Early humans need not have been students of formal geometry to know and to use these ideas.

As early as 2,600 years ago the Greeks had not only discovered a large number of geometric properties, they had begun to see them as abstract ideas to be studied in their own right. By the third century B.C., they had created a formal system of geometry. Their system began with the simplest ideas and, with these ideas as a foundation, went well beyond much of what is taught in schools today.

## Additional Topics

Typically one learns arithmetic and algebra by experiment or by being told how to do it. Geometry, however, is taught logically. Its ideas are established by means of "proof." One starts with definitions, postulates, and primitive terms; then proves his or her way through the course. The reason for this goes back to the forenamed Greeks, and in particular to Euclid. Twenty-three hund…

Another lasting influence of Euclid's Elements is the emphasis which is placed on constructions. Three of the five postulates on which Euclid based his geometry describe simple drawings and the conditions under which they can be made. One such drawing (construction) is the circle. It can be drawn if one knows where its center and one point on it are. Another construction is drawing a line s…

Points, lines, and planes are primitive terms; no attempt is made to define them. They do have properties, however, which can be explicitly described. Among the most important of these properties are the following: Two distinct points determine exactly one line. That line is the shortest path between the two points. Bricklayers use these properties when they stretch a string from corner to corner …

An angle in geometry is the union of two rays with a common endpoint. The common endpoint is called the "vertex" and the rays are called the "sides." Angle ABC is the union of BA and BC. When there is no danger of confusion, an angle can be named by its vertex alone. It is also handy from time to time to name an angle with a letter or number written in the interior of t…

Given a line and a point not on the line, there is exactly one line through the point parallel to the line. These principles are used in a variety of ways. A draftsman uses 2) to rule a set of parallel lines. Number 1) is used to show that the sum of the angles of a triangle is equal to a straight angle. If a set of parallel lines cuts off equal segments on one transversal, it cuts off equal segme…

If A is a given point and CD a given line, then there is exactly one line running through A that is perpendicular to CD. If B is the point on line CD that also resides on the line running perpedicular to CD, then that line, AB, is the shortest distance from point A to line CD. In a plane, if CD is a line and B a point on CD, then there is exactly one line through B perpendicular to CD. If B happen…

Triangles are plane figures determined by three non-collinear points called "vertices." They are made up of the segments, called sides, which join them. Although the sides are segments rather than rays, each pair of them makes up one of the triangle's angles. Triangles may be classified by the size of their angles or by the lengths of their sides. Triangles whose angles are al…

A circle is a set of points in a plane which are a fixed distance from a point called the center, C (see Figure 4). A "chord" is a segment, DE, joining two points on the circle; a radius is segment, CA, joining the center and a point on the circle; a diameter is a chord, DB, through the center. A "tangent," DF, is a line touching the circle in a single point. The words …

The area of a quadrilateral with sides a, b, c, and d depends not only on the lengths of the sides but on the size of its angles. When the quadrilateral is cyclic (all four endpoints are on in a circle), its area is given by a remarkable formula discovered by the Hindu mathematician Brahmagupta in the seventh century:
where s is the semi-perimeter (a + b +c + d)/2. This formula includes Hero…

The foregoing is a summary of Euclidean geometry, based on Euclid's postulates. Euclid's fifth postulate is equivalent to assuming that through a given point not on a given line, there is exactly one line parallel to the given line. When one assumes that there is no such line, elliptical geometry emerges. When one assumes that there is more than one such line, the result is hyperboli…

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