Plain Plane Geometry

The following sections are included:

• Dedication
• Preface
• Contents

In this chapter, we will introduce some of the basic geometric characters in our geometric “play” to follow in this book. In some sense, this is the most boring chapter of the book, as it does not have much “action”. The real fun in geometry begins when things start interacting. So a certain amount of patience is required. But this is true when learning any new game. One needs to know how the chess pieces move before one can start playing the game, and that part of learning the rules is not very exciting. It is the same with geometry…

Two figures which can be obtained from each other by a rigid motion (rotation, translation or reflection) are considered to be the “same” or congruent in planar geometry. Indeed they have the same shape and size. For example, line segments of equal lengths are congruent, angles of equal measure are congruent, circles of the same radius are congruent. In this chapter, we will study congruency of triangles…

Definition 3.1 (Quadrilateral). A quadrilateral is a figure with 4 sides and 4 vertices.

Loosely speaking, a side is a line segment subset of the planar figure which is maximally contained in the figure; in other words, if we add any more points to the subset by extending the line segment, then we fall outside the figure. A vertex is a point where two distinct sides of the figure meet…

Two figures in the plane are congruent if by a rigid transformation (reflection, translation or rotation), one figure can be transformed into the other. Thus congruent triangles have the same shape and size. But now we will look at a more general type of transformation which also allows “scaling”. Imagine looking at a portion of the plane through a magnifying glass for example. The image you see has the same shape as what is actually there on the page, except that its size has been “scaled” (magnified). Then we say that the resulting image is similar to the original image. For example, in the pictures below, we see scaled versions of the square by factors of 2, 1/2, , and each of the resulting squares is similar to each of the others…

We had defined the terms circle, its center and its radius in Section 1.6. In this chapter, we will study circles. We begin with some terminology. Recall that a circle C(O, r) with center O and radius r > 0 is the set of points P in the plane such that OP = r. The set of points P in the plane such that OP < r is called the interior of the circle, while the set of points P for which OP > r is called the exterior of the circle. Circles having the same center are said to be concentric…

Let us make a few concluding remarks here about what we have learnt, what has not been done, and where the reader might wish to go from here…

The following sections are included:

• Hints to the exercises from Chapter 1
• Hints to the exercises from Chapter 2
• Hints to the exercises from Chapter 3
• Hints to the exercises from Chapter 4
• Hints to the exercises from Chapter 5

The following sections are included:

• Solutions to the exercises from Chapter 1
• Solutions to the exercises from Chapter 2
• Solutions to the exercises from Chapter 3
• Solutions to the exercises from Chapter 4
• Solutions to the exercises from Chapter 5
• Notes

The following sections are included:

• Bibliography
• Index