1. 1.2 Points, Lines and Planes Wherein we define the fundamental concepts of geometry – point, line, plane, on, between and distance. This is our basic set. We’ll use them to define all other terms. Textbook Section 1-1

2. Definitions When we begin any mathematical inquiry, we must first offer precise definitions of our terms. So let us begin with definitions of the basic concepts of Euclidean Geometry. These are point, line, plane, on, between and distance. All other terms will defined in terms of this basic set.

3. Point Defined A point is the smallest of our objects. It has no length, width or depth and thus is 0- dimensional. Points thus have no parts. They are atoms in the original sense of that term. The best model of a point is a corner. All other objects – lines, planes, angles, circles and all the rest – are made of points.

4. How to Name and Draw Points Points are represented by dots. Capital letters name them. Illustration:

5. Line Defined A line has no width or depth, but it does have length. Thus we say that lines are 1-dimensional. A line is straight, i.e. it does not curve. A line is continuous, i.e. it has no gaps. A line is infinite, i.e. it has no end-points. Our best model of a line is an edge.

6. How to Name and Draw Lines We will find that at most one line can pass through a pair of points. (More on this later.) Thus we may name a line by a pair of capital letters, each of which names a point on the line. Write them one after the other and place a little line on top. Lower-case script letters also serve as names of lines.

7. Illustration

8. Plane Defined Planes have no depth but they do have length and width. We thus say that planes are 2- dimensional. Planes are flat, i.e. they do not curve. Planes are continuous, i.e. they have no gaps. Planes are infinite in both length and width. Our best physical model of a plane is the surface of a flat object.

9. How to Draw and Label Planes We will find that at most one plane can pass through three points not all on a line. (As before, we’ll take this up later.) Thus we may name planes with sets of three capital letters. Write them one after the other. Write the word ‘plane’ in front. We may also name planes with upper-case script letters.

10. Illustration

11. On Defined

12. Between Defined If as we pass along a line, we encounter first one point, then a second and finally a third, we say that the second is between the first and the third. Thus a point is between two others only when all lie on a line; and when three points lie on a line, one is between the other two.

13. Illustration Point M lies between points A and B.

14. Distance Defined If two points lie on a line, we say that there is a positive real number that represents the distance between them. We thus associate pieces of lines (later we’ll call them segments) with positive reals. The so-called Ruler Postulate will tell us how to compute distance. It will be stated tomorrow.

15. The Basic Set We now have a stock of basic terms on which to draw. They are: point, line, plane, on, between and distance. All others (with one exception in Chp. 3) will be defined in terms of them. We’ll end today with the definitions of the terms figure, collinear, coplanar and intersection. In each we’ll make use of the terms of our basic set and those terms alone.

16. Figure Defined We need a generic term for all of the geometrical objects we’ll study. The term is figure. A figure is simply a collection one or more points. Points are figures. Lines are figures. Planes are figures. Everything in this course is a figure.

17. Collinear and Coplanar Defined

18. Intersection Defined Figures intersect when one or more points are on both. The intersection of two figures is the set of points on both. Students often assume that the intersection of two figures is only a part of each. This is a mistake. The intersection of two figures can be a part of each. But it can also be the whole of one and a part of the other.

19. Example What is the intersection of two lines?

20. Another What is the intersection of two planes?

21. A Third What is the intersection of a line and a plane?

22. Draw and Label