Introduction
Geometry is full of surprises. For instance in any triangle, the altitudes drawn from the vertices to the opposite sides all meet in a single point, the orthocenter of the triangle. It's always interesting when three or more lines, like the three altitudes of a triangle, meet at a point or when three or more points lie on a line. "Seeing is believing" removes some of the mystery associated with these geometrical surprises, and sheds some light on why they happen. Dynamic geometrical software, such as The Geometer's Sketchpad, is more than just a computerized compass and straightedge. Like a spreadsheet, it maintains relationships among constructed objects so that you can see what things remain the same and what things vary as others change. It's fairly simple to illustrate that the three altitudes of a triangle always meet in a single point using Sketchpad - construct a triangle, its altitudes and drag the vertices of the triangle around; the altitudes will always meet in a point.
The following activities in dynamic geometry use The Geometer's Sketchpad to explore surprising and interesting things about triangles, circles, polygons, symmetry, conic sections and hyperbolic geometry. Some of these are well known results, such as the sum of the angles in a triangle is 180°, the Pythagorean Theorem, and the inscribed angle theorem for circles, which states that an inscribed angle is half the size of its intercepted arc. The last result provided the key to understanding the two chord theorem, the fact that two chords of a circle divide each other into segments, the product of whose lengths is constant, and this in turn leads directly to the idea of the radical axis of two circles and circular inversion, an important but not well known transformation of the plane that leads to lots of surprises. Some other surprises that are illustrated here are the limited number of possible "wallpaper patterns", i.e., the limited number of ways to repeat patterns in the plane, and the focal (or reflective) properties of the conic sections - parabolas, ellipses and hyperbolas - that make them so important in eyeglasses, telescopes and all sorts of other equipment involving light and sound. Surprises abound in hyperbolic geometry -no longer are parallel lines unique, no longer is the angle sum of a triangle always 180°, no longer is the Pythagorean Theorem the familiar a²+b²=c². There are also remarkable and strange ways to tile the hyperbolic plane with regular polygons, for example with squares five to a vertex!
The activities that follow lead you through many of these interesting and surprising results. Often the true beauty of a result lies in it "proof", the justification of the result. (It is interesting to note that Gauss, one of the most famous mathematicians was incredibly proud of his proof of the constructability of a regular 17-gon with just a compass and straightedge, but never actually carried through the construction on paper.) As a general rule, proofs are not included here, but some of the ideas behind the proofs are presented in the questions accompanying the activities. There are more than 700 questions that encourage you to try variations and extensions of each activity on your own.
Seeing may be believing, but it can make a result seem obvious or less interesting than it actually is. There are many fancy interactive web pages that illustrate some of the results you will construct in the following pages, and you might want to explore some of them, but I believe there is merit in constructing your own illustrations, rather than just looking at the final product of someone else's work. There is something to the saying that one learns by doing. Among other things, this should stimulate your curiosity about why a certain result holds, and how to argue that it does.
There are many topics of geometry not represented here. Maybe these activities will lead you to explore some of them on your own. For example, coordinate geometry in the xy-plane, what some people nowadays identify with geometry as a whole makes only a brief appearance here. There are also many other types of geometries not included here, from finite geometries to elliptic and projective geometries, two fascinating geometries in which there are not parallel lines at all!
This is also not a tutorial on how to use The Geometer's Sketchpad. The reference manual and other materials from Key Curriculum Press provide a wealth of ideas about how to make sketches more exciting and dynamic. Nevertheless, you will learn things about animating sketches, making action buttons and creating custom tools as you work through these activities. Even more, I hope these activities encourage your interest in and appreciation of geometry.