Über die Autorin bzw. den Autor

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, New York. Previously, he had the same positions at the City College of the City University of New York for forty years.  He has published over fifty-five books in the area of mathematics and mathematics education, including The Fabulous Fibonacci Numbers (with Ingmar Lehmann). 
 
Ingmar Lehmann is retired from the mathematics faculty at Humboldt University in Berlin. For many years he led the Berlin Mathematics Student Society for gifted secondary-school students, with which he is still closely engaged today.  He is the coauthor with Alfred S. Posamentier of The Secrets of Triangles, The Glorious Golden Ratio, and three other books.

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

THE SECRETS OF TRIANGLES

Prometheus Books

Contents

Acknowledgments.........................................9Preface.................................................11. Introduction to the Triangle.........................152. Concurrencies of a Triangle..........................333. Noteworthy Points in a Triangle......................654. Concurrent Circles of a Triangle.....................895. Special Lines of a Triangle..........................1056. Useful Triangle Theorems.............................1357. Areas of and within Triangles........................1618. Triangle Constructions...............................1979. Inequalities in a Triangles..........................25710. Triangles and Fractals..............................293Appendix................................................335Notes...................................................359References..............................................367Index...................................................369

Chapter One

Arithmetic! Algebra! Geometry! Grandiose trinity, brilliant triangle! Who has not known you, is a poor wretch! ... But who knows you and appreciates you, desires no further goods of the earth. —The Songs of Maldoror II, 10

The word triangle is used in a variety of contexts. For example, there is the Bermuda Triangle, which refers to the area of a triangle determined by three points: one at Miami, Florida; another at San Juan, Puerto Rico; and a third at Bermuda. It is believed that this triangular surface has had an inordinate number of ship and aircraft mishaps. There is also another well-known triangular region called the Summer Triangle: three stars that determine a triangle. The summer triangle consists of the stars known as Deneb, Altair, and bluish Vega. The American essayist Henry David Thoreau (1817–1862) has been often quoted with the following: "The stars are the apexes of what triangles!"

Then there is the culinary triangle, a concept described by anthropologist Claude Lévi-Strauss (1908–2009) involving three types of cooking; these are boiling, roasting, and smoking, usually done to meat. Here the triangle is determined by the three sides or angles, depending on how it is used. Then there is the social triangle as described by the French writer Honoré de Balzac (1799–1850). The three points of the social triangle are skill, knowledge, and capital. Another triangle determined by the three sides is the musical instrument the triangle. What we then have is a variety of ways that we can define a triangle geometrically: either a polygon of three sides, or three noncollinear points, or the area within the region determined by the previous two definitions.

The triangle is the basic geometrical figure that allows us to best study geometrical shapes. A quadrilateral can be partitioned into two triangles, a pentagon into three triangles, a hexagon into four triangles, and so on. (See figures 1-1a, 1-1b, and 1-1c.) These partitions allow us to study the characteristics of these figures. And so it is with Euclidean geometry—the triangle is one of the very basic parts on which most other figures depend.

Yet before we embark on our journey investigating triangles and their many related line segments and angles, we ought to determine what it takes for a triangle to exist. Suppose you have three rods and the sum of the lengths of two of them is shorter than the length of the third rod, then you will see that you cannot form a triangle with the three rods. (See figure 1-2.)

We can generalize this by saying that in order for a triangle to exist, the sum of the lengths of any two sides must be greater than the third side.

TRIANGLE CONGRUENCE

Let us now review the various relationships that can connect two triangles. First there is the congruence of two triangles (using the symbol [congruent to]), which describes two triangles with the exact same size and shape so that they can be placed to perfectly coincide. In other words, the corresponding sides and angles of the two triangles are equal. To show that two triangles are congruent, we do not need to determine that all the corresponding sides and angles are equal. Rather we can establish the congruence of two triangles simply by showing that any one of the following is true:

• The three sides of one triangle (?ABC) are equal to the three corresponding sides of the other triangle (?DEF).

• Two right triangles can be shown to be congruent if the hypotenuse and a leg of one triangle are equal to the corresponding sides of the second triangle.

• Two sides and the angle between them of one triangle (?ABC) are equal to corresponding parts of the other triangle (?DEF). (See figure 1-4.)

• Two angles and one side of one triangle (?ABC) are equal to the corresponding parts of the other triangle (?DEF).

We indicate this congruence symbolically as ?ABC [congruent to] ?DEF.

Another relationship between two triangles is similarity (represented by the symbol ~), which tells us that the two triangles have the same shape but not necessarily the same size, that is, that the corresponding angles of the two triangles are equal. Similarity between two triangles can be established by showing that:

• Two angles of one triangle (?[ABC) are equal to two angles of the other triangle (?DEF) as shown in figure 1-5.

• The three sides of one triangle (?ABC) are proportional to the three sides of the other triangle (?DEF).

• Two sides of one triangle (?ABC) are proportional to two sides of the other triangle (?DEF) and the angles between these two sides of each triangle are congruent.

Symbolically we write this as ?ABC ~ ?DEF .

Two triangles can also be related by their position in the plane. For example, consider two triangles, ?ABC and ?A'B'C ( of possibly different shapes), whose corresponding sides (extended) meet in three collinear points X, Y, and Z (i.e., points that lie on the same straight line):

sides AC and A'C' meet at point X, sides BC and B'C' meet at point Y, and sides AB and A'B' meet at point Z.

Then the lines joining the corresponding vertices (AA', BB', and CC' ) are concurrent (in point P), as shown in figure 1-6. This famous twotriangle relationship was first discovered by the French mathematician and engineer Gérard Desargues (1591–1661) and today bears his name. The converse of this relationship is also true. Namely, if two triangles are so placed that the lines joining their corresponding vertices are concurrent (in figure 1-6, point P is that point of concurrency), then the extensions of their corresponding sides will meet in three collinear points (points X, Y, and Z).

THE EQUILATERAL TRIANGLE

There are also triangles that have special relationships within themselves. Perhaps the most common is the equilateral triangle, which is one that has all sides equal and all angles equal. Not only that, but all of its angle bisectors, altitudes, and medians are equal to each other. A lesser-known...