Points, Lines and Planes are undefined terms in Geometry whose meaning is accepted without definition. These terms have similar meanings in Geometry as they are used in algebra, like the coordinate plane, ordered pairs and lines . Let us see the description of these terms and how they are represented.

Description
Representation Diagram
A represents just a location and does not have a size.
 A point is represented by a dot and is named
using a capital letter. The Point is A is shown here.
 Point Introduction
 A line has no thickness and contains infinitely many points and it represents a straight path
extending in opposite directions.
 A line is represented by two points on the line,
like Line Introductionor What is Line. A line is also represented by a single lower case
letters like 'l'.
 Line Introduction
 A plane is a flat surface made up of infinitely many points extending in all direction without
thickness.
A plane can be represented by a capital letter like Plane X.
It can also be represented by three points
which do not lie on the same line like
Plane ABC.
 Plane Introduction


Now, let us learn about some special points, lines and planes.

Collinear Points: A collinear set of points is a set of points all of which lie on the same straight line.
Collinear Points

In the above diagram, the points A, B. C and D are all points lying on the line 'l'.

Two points are sufficient to determine a line. Hence we discuss collinearity of three or more points and not with two points.

Non-Collinear Points: A non collionear set of points is a set of three or more points that do not all lie on the same straight line.
Non-Collinear Points

Three non collinear points define a plane and they form the vertices of a triangle.
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Two rays with a common end point make an angle at the common point. The common point is called the vertex of the angle.

Vertex

Polygons are formed by intersecting lines. The shape of a polygon is determined by the number of sides and the angles they make.
The points of intersections of the adjacent sides are also known as the vertices (plural of vertex) of the polygon.
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When three or more straight lines intersect to have a common point, the lines are said to be concurrent at the common point. The common point is known as the point of concurrency of the lines.

Point of Concurrency

Two non parallel lines, definitely concur at a common point. But if three are more lines concur at a point it is a special phenomena.
We learn four points of concurrency related to triangles.

Circumcenter is the point of concurrency of the perpendicular bisectors of the three sides of a triangle.

Incenter is the point of concurrency of the angle bisectors of the three angles of a triangle.

Orthocenter is the point of concurrency of the three altitudes of a triangle.

Centrroid or Center of gravity is the point of concurrency of the three medians of a triangle.

The distance between points between a point and a line or the distance between a point and a plane can be found. Equidistant points can be also found from a point, from a line or plane can be also found.

Examples:
1. The mid point of a line segment joining two points is equidistant from the two end points.

Mid Point

2. Any point on the perpendicular bisector of a line segment is equidistant from the end points.
3. The center of a circle is a point equidistant from all the points lying on the circumference of the circle.

     Circle

    The equal distance is the radius of the circle.
4. A set of points equidistant from a plane form a plane parallel to the given plane.
Locus is a set of points satisfying a given condition.

Locus can also be viewed as the path traced by a moving point subject to certain conditions.
We saw in the example for equidistant points, that any point on the perpendicular bisector of a line segment is equidistant from the end points of the line segment. It can also be proved that any point equidistant from two given points lie on the perpendicular bisector of the line segment joining the two points. Thus we can say the perpendicular bisector of a line segment is the locus of the points which are equidistant from the end points of the line segment.

Perpendicular Bisector

Similarly a circle is the locus of points which all lie at a given distance from a given point.

The pair of angle bisectors of two intersecting lines is the locus of all points which are equidistant from the two lines.

Angle Bisector

A ray is a part of a straight line. It has one end point and extends infinitely in the opposite direction.  Rays are represented starting with the end point first and any other point on the ray like $\overrightarrow{AB}$, where A is the end point.
Ray
 
A point on a line determines exactly two rays called opposite rays.

Opposite Rays

In the above diagram $\overrightarrow{PQ}$   and $\overrightarrow{PR}$ are opposite rays with the common end point P.
Line segments are parts of the lines which are terminated at both the ends by the end points. As shown in the earlier examples  Line Segmentsis the line segment with end points A and B. It can also be indicated by Line Introduction.

The midpoint of a line segment is a point equidistant from the end points.
Mid Point

In the above picture C is the midpoint of Line Segment, which is equidistant from both A and B. → Read More
Two lines are called intersecting lines, if they cross each other at a common point.
Intersecting Lines

Vertical angles formed by two intersecting lines are congruent. In the above diagram lines 'l' and 'm' intersect at the common point 'O'. Vertical angles 1 and 3 are congruent. Similarly the other pair of vertical angles 3 and 4 are also congruent.
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Coplanar lines which do not intersect, however much they are extended on either side are called the parallel lines.
The symbol || is used to indicate two lines are parallel. AB || CD reads as Line AB is parallel to line CD. In a diagram, the two lines are marked with arrows in the same direction to indicate that they are parallel.

Parallel Lines

A parallelogram gets its name from the fact that its opposite sides are parallel.

If two lines are parallel, then their slopes are equal.
Lines that form right angles are perpendicular lines. When two perpendicular lines intersect they form four right angles.

If two lines are perpendicular to each other, then the product of their slope = -1. That is if m1 and m2 are the slopes of two perpendicular lines, then m1 x m2 = -1.
In coordinate plane, the lines parallel to the horizontal axis or the x-a xis are called the horizontal lines.

Horizontal Lines
The equation to a horizontal line is of the form y = k where k is the signed distance between the line and the x axis. The slope of horizontal lines are zero.
Vertical lines are parallel to the vertical axis, Y-axis. Thus vertical lines are perpendicular to the horizontal axis and horizontal lines.

Vertical Lines

The equation to a vertical line is of the form x = k, where k is the signed distance between the line and the y axis. The slope of the vertical lines are undefined similar to the slope of y-axis.
A plane is a set of points that form a flat surface extending in all directions. The surface of the floor of a room or the surface of your writing table are examples for plane. A Plane can be represented by a single capital letter or at least three non collinear points on the plane.
Definition of Plane

If a plane can be found containing a set of given points, then the points are said to be coplanar.

Similarly a set of lines are contained in a plane, then the lines are called coplanar.

Same way a set of shapes like triangles are called coplanar, if they all fall in a common plane.
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