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THE SECRETS OF TRIANGLES
A MATHEMATICAL JOURNEY
By ALFRED S. POSAMENTIER INGMAR LEHMANN
Copyright © 2012 Alfred S. Posamentier and Ingmar Lehmann
All right reserved.
Chapter One INTRODUCTION TO THE TRIANGLE
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.
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 (ΔABC) are equal to the three corresponding sides of the other triangle (ΔDEF).
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 (ΔABC) are equal to corresponding parts of the other triangle (ΔDEF). (See figure 1-4.)
Two angles and one side of one triangle (ΔABC) are equal to the corresponding parts of the other triangle (ΔDEF).
We indicate this congruence symbolically as ΔABC [congruent to] ΔDEF.
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 (Δ[ABC) are equal to two angles of the other triangle (ΔDEF) as shown in figure 1-5.
The three sides of one triangle (ΔABC) are proportional to the three sides of the other triangle (ΔDEF).
Two sides of one triangle (ΔABC) are proportional to two sides of the other triangle (ΔDEF) and the angles between these two sides of each triangle are congruent.
Symbolically we write this as ΔABC ~ ΔDEF .
Two triangles can also be related by their position in the plane. For example, consider two triangles, ΔABC and ΔA'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 property of the equilateral triangle is seen by taking any point, P, in an equilateral triangle (figure 1-7) and drawing the perpendicular segments to each of its three sides. The sum of the distances from this randomly chosen point to the three sides, PQ + PR + PS, is always the same. That sum is equal to the altitude of the equilateral triangle. This is shown in figure 1-7, where the altitude is CD. This relationship, often called Viviani's theorem, is attributed to the Italian mathematician Vincenzo Viviani (1622–1703), who, incidentally, was a student of the famous Italian scientist and philosopher Galileo Galilei (1564–1642).
This surprising property can be proven by using the formula for the area of a triangle (i.e., the area of a triangle is one-half the product of the base and the altitude drawn to that base). We begin with equilateral triangle ABC, where PR [perpendicular to] BC, PQ [perpendicular to] AB, PS [perpendicular to] AC, and CD [perpendicular to] AB. We then draw PA, PB, and PC, as shown in figure 1-8.
The Area ΔABC = Area ΔAPB + AreaΔBPC + AreaΔCPA = 1/2 AB · PQ + 1/2 BC · PR + 2 AC · PS.
Since AB = BC = AC, the AreaΔABC = 1/2 AB · (PQ + PR+ PS).
However, the AreaΔABC = 1/2 AB · CD. Therefore, PQ + PR+ PS= CD, which is then a constant that we sought to prove true for the given triangle.
There are many other relationships special to the equilateral triangle beyond the basic ones we just mentioned. The surprising properties of the equilateral triangle will be presented a bit later. In the meantime, we shall survey some other special triangles. The isosceles triangle is one that has at least two sides of the same length. Its base angles are always equal to each other. We will be revisiting the isosceles triangle in our discussions throughout the book.
THE RIGHT TRIANGLE
The right triangle is so named because it has one right angle, as shown in figure 1-9, where [angle]ACB = 90°.
It, too, has many properties within itself. For example, when an altitude is drawn to the hypotenuse of the right triangle, the triangle is partitioned into three similar triangles. In figure 1-10, the three similar triangles are ΔABC ~ ΔACD ~ ΔBCD.
If we look at these three triangles in pairs, we can establish a rather familiar relationship. Follow along!
We will begin with ΔABC ~ ΔACD. From this similarity we get the following proportion of their side lengths: AB/AC = AC/AD. This gives us AC2 = AB · AD. From the similarity ΔABC ~ DBCD, we get AB/BC = BC/BD , or BC2 = AB · BD. When we add these two equations, the following results: AC2 + BC2 = AB · (AD + BD) = AB2. When we express this verbally, we have the following statement: "The sum of the squares of the legs of a right triangle equals the square of the hypotenuse."
This should remind us of perhaps the most famous theorem in geometry, the Pythagorean theorem. If we replace "of" with "on" in this statement, we have, referring to the areas of the squares, "the sum of the squares on the legs of a right triangle is equal to the square on the hypotenuse.
This can then be shown geometrically as in figure 1-11, namely, the sum of the areas of the two smaller squares (those on the legs of the right triangle) is equal to the area of the larger square—the one on the hypotenuse.
We know that any three noncollinear points determine a unique triangle as well as a unique circle. However, when the triangle is a right triangle, then the circumscribed circle's diameter is the hypotenuse of the triangle, as shown in figure 1-12, where hypotenuse AB is the diameter of the circumscribed circle c with the midpoint O.
From this property, we can show very easily that if we draw the median to the hypotenuse of a right triangle, we will have formed two isosceles triangles, ΔAOC and ΔBOC. We show this in figure 1-13, where CO is the median to the hypotenuse of right triangle ABC, but in this case, the median—which is also the radius of the circumscribed circle—is half the length of the hypotenuse. Therefore, CO = BO = AO. Consequently, ΔAOC and ΔBOC are isosceles triangles.
As we mentioned earlier, just as the right triangle is categorized by one of its angles—the angle of 90-degree measure—other triangles can also be categorized by a triangle's angle measures. When a triangle has an angle greater than 90°, which is called an obtuse angle, the triangle is called an obtuse triangle. When all of a triangle's angle measures are each less than 90° (i.e., acute angles), then we call the triangle an acute triangle.
An extension of the Pythagorean theorem allows us to establish relationships among the sides of a triangle that will help us to determine if a nonright triangle is acute or obtuse.
For a triangle whose sides have lengths a, b, and c, if a2 + b2 > c2, then the angle between the sides of length a and b is acute (see figure 1-14) and the triangle is an acute triangle.
On the other hand, if a triangle's sides have lengths a, b, and c, and if a2 + b2 < c2, then the angle between the sides of length a and b is obtuse (see figure 1-15), and the triangle is then an obtuse triangle.
Moreover, for an obtuse triangle, such as ΔABC, shown in figure 1-16, we have the following relationship, which derives directly from the Pythagorean theorem: c2 = a2 + b2 + 2ax. In other words, this would make c2 greater than a2 + b2, which we stated before.
Now, to show why the equation c2 = a2 + b2 + 2ax is true. Using figure 1-16, and applying the Pythagorean theorem (first to triangle ABD): c2 = (a + x)2 + h2 = a2 + x2 + 2ax + h2 = a2 + (x2 + h2) + 2ax. However, applying the Pythagorean theorem to triangle ADC, we get b2 = x2 + h2. Therefore, c2 = a2 + b2 + 2ax, which is what we set out to show above.
For an acute triangle, shown in figure 1-17, we have c2 = a2 + b2 - 2ax. This can be veriied in a manner similar to the method used above, and would justify that c2 is less than a2 + b2, which we also stated before.
The Pythagorean theorem allows us to arrive at lots of interesting triangle relationships. For example, there is one that is attributed to Apollonius of Perga (ca. 262–ca. 190 BCE), which states that for triangle ABC, with median AD, we can show that AB2 + AC2 = 2 (BD2 + AD2). (See figure 1-18.)
We can determine the area of a triangle in a number of ways depending on the information given about the triangle. If we are given the length of one side of the triangle and the length of the altitude drawn to that side, then we can use our familiar formula for the area: one-half the product of the base and its height. Symbolically that is written as Area = 1/2 bh. (See figure 1-19.)
If we are given the measure of one angle, for example, [angle]A = α, of a triangle ABC and the lengths of the two sides forming that angle, b and c, then we have the following additional formula for the area of triangle ABC. Symbolically that is written as Area ΔABC = 1/2 bc · sin [angle]A = 1/2 bc · sin α.
It is also possible to establish the area of a triangle given the lengths a, b, and c, of the three sides of triangle ABC using Heron's formula Area ΔABC = [square root of s(s - a)(s - b)(s - c)], where s = 1/2 (a + b + c) is the semi-perimeter of the triangle ABC.
We will be exploring the area of triangles and other related areas in chapter 7.
TRIGONOMETRY AND THE TRIANGLE
The Pythagorean theorem is actually the basis for all of trigonometry, therefore, of the over four hundred proofs of the Pythagorean theorem that exist today, none uses trigonometry—or else we would have circular reasoning. (Remember, you cannot prove a theorem using a relationship that depends on that theorem!) Yet, with the advent of trigonometry, we have some very useful relationships surrounding the triangle. Each named after one of the three basic trigonometric functions: sine, cosine, and tangent.
Let us first review these basic functions as they apply to a right triangle, and then provide their application to the general triangle. For the right triangle, ΔABC (figure 1-20), we have three trigonometric functions defined for [angle]A as follows:
sin [angle]A = a/c
cos [angle]A = b/c =
tan [angle]A = a/b
Extending these trigonometric functions to the general triangle we get the following relationships, known as the law of sines:
a/sin [angle]A = b/sin [angle]B = c/sin [angleITL
It is interesting to see how easily this relationship evolves from the basic above-mentioned definitions of the sine function. To begin, we will consider triangle ABC, with altitude CD (= hc) to side AB (= c), which partitions the triangle into two right triangles, and ΔACD ΔBCD. (See figure 1-21.)
In the right triangles ΔACD and ΔBCD, we can apply the sine function as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Therefore, hc = b sin [angle]A = a sin [angle]B, or a/sin [angle]A = b/sin [angle]B. (We can also write this as a/sinα = b/sinß.)
Excerpted from THE SECRETS OF TRIANGLES by ALFRED S. POSAMENTIER INGMAR LEHMANN Copyright © 2012 by Alfred S. Posamentier and Ingmar Lehmann. Excerpted by permission of Prometheus Books. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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