Triangle Inequalities

4 Key excerpts on "Triangle Inequalities"

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  Inequalities

  Theory of Majorization and Its Applications

  • Ingram Olkin, Albert W. Marshall(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  This unification also has the advantage of suggesting new inequalities. Because of the repeated references made to it, the book Geometric In-equalities, cited above, is referred to as G.I. Other inequalities are from the American Mathematical Monthly, which is referred to more simply as the Monthly. References where the dates appear without parentheses are taken from G.I. and are not repeated in the bibliography of this book. That majorization can play a role in generating inequalities for the tri-angle was noted by Steinig (1965), who obtained majorizations between the sides of a triangle and the exradii, and between the sides of a triangle and the medians. For the triangle, the sum of the angles is fixed, and majoriza-tion arises quite naturally. Many inequalities for the angles of a triangle are obtained as a direct application of a majorization using a Schur-convex function. Subsequent to the preparation of this chapter, a paper by Oppenheim (1971) was published in 1978 that also contains the idea of using majoriza-tion to obtain inequalities for the triangle. 192 Sect. A INEQUALITIES FOR THE ANGLES OF A TRIANGLE 193 In presenting these inequalities, the case of equality (when it can be achieved) is often readily identified. The reason for this is that if φ is strictly Schur-convex, then in an inequality of the form φ(χ) > φ(α) for all x >■ a, or of the form φ(χ) < φ(α) for all x -< a, equality holds only if x is a permutation of a. The Schur-convex functions used in this chapter are strictly Schur-convex (mostly as a consequence of 3.C.l.a). Trigonometric inequalities for the triangle comprise Section A and are organized according to whether they relate to the sines, cosines, or tangents of the angles of a triangle. Other inequalities for the cotangents, secants, and consecants of the angles are obtainable, but are omitted because they follow similar patterns and tend to be repetitive.

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  College Geometry

  A Unified Development

  • David C. Kay(Author)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  281 6 Inequalities.for. Quadrilaterals:.Unified. Trigonometry In order to advance our knowledge of the two non-Euclidean geometries, it is necessary to develop a non-Euclidean “trigonometry.” This necessi-tates the development of a few inequalities for midlines in right triangles and Lambert quadrilaterals. That will be the goal of the first part of this chapter. The development of unified trigonometry will then occupy the second half—material that includes a special “Pythagorean theorem” for the two non-Euclidean geometries. 6 .1. An.Inequality.Concept.for.Unified.Geometry One of the major, defining concepts for unified geometry involves the angle-sum for triangles. In elliptic geometry, the angle-sum of a triangle is greater than 180, while in hyperbolic geometry, the angle-sum is less than 180 (and in Euclidean geometry, of course, it is 180). It is convenient at this point to introduce a special type of inequality that will apply to all three geometries simultaneously. It might be referred to as a unified inequality . While it does not necessarily simplify proofs (more often than not, proofs must be constructed separately for the three geometries), it does save in writing and in stating the inequalities concisely. The idea is this: We let a * b denote the relation in unified geometry that represents a < b , a = b , or a > b , respectively, in hyperbolic, Euclidean, or elliptic geometry. Instead of the asterisk, a symbol more suggestive of inequality will be used: a “curved” inequality symbol, to distinguish this from ordi-nary inequality. The formal definition is then Definition If a and b are real numbers, write a ≺ b iff a < b in hyperbolic geometry, a = b in Euclidean geometry, or a > b in elliptic geometry. Define a ≻ b iff b ≺ a . It is immediately clear that this new relation (inequality) has all the attributes of ordinary inequality for real numbers, since it coincides with

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  Selected Topics in Geometry with Classical vs. Computer Proving

  • Pavel Pech(Author)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  This is based on the expression of a given inequality in terms of other geometric magni-tudes from which the inequality is seen. Using this method the inequality of Euler is solved. Besides computer proofs we shall show classical solutions as well, to compare both approaches. 163 164 Selected Topics in Geometry with Classical vs. Computer Proving 7.1 Inequality between the diagonals of an n -gon The best known geometric inequality is probably the triangle inequality which says that the sum of two sides of a triangle is greater than the third side. We will deal with the inequality between the sides and diagonals of an n -gon. To simplify the situation we will use the word “diagonal” for any segment joining two vertices of an n -gon. First look at a quadrilateral. We start with an investigation of equalities between diagonals holding for various types of quadrilaterals. 7.1.1 Parallelogram law In this part we will explore the equality between the sum of squares of sides and the sum of squares of diagonals of a parallelogram. This equality is known as the parallelogram law [44]: Theorem 7.1 (Parallelogram law). Given a parallelogram with the side lengths a, b and diagonals e, f . Then 2( a 2 + b 2 ) = e 2 + f 2 . (7.1) Let us prove the relation (7.1). Denote the vertices of a parallelogram by the letters A, B, C, D and its side lengths and diagonals by a = | AB | = | CD | , b = | BC | = | DA | , e = | AC | , f = | BD | . Choose a Cartesian coordinate x y b a e f A=[0,0] B=[a,0] C=[x,y] D=[x-a,y] j Fig. 7.1 Parallelogram law: 2( a 2 + b 2 ) = e 2 + f 2 system such that A = [0 , 0] , B = [ a, 0] , C = [ x, y ] , D = [ x − a, y ] (Fig. 7.1). Then Geometric inequalities 165 b = | BC | = | AD | ⇔ h 1 : ( x − a ) 2 + y 2 − b 2 = 0 , e = | AC | ⇔ h 2 : x 2 + y 2 − e 2 = 0 , f = | BD | ⇔ h 3 : ( x − 2 a ) 2 + y 2 − f 2 = 0 . The conclusion c of a statement has the form c : 2( a 2 + b 2 ) − ( e 2 + f 2 ) = 0 .

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  Dr. Math Presents More Geometry

  Learning Geometry is Easy! Just Ask Dr. Math

  • (Author)
  • 2005(Publication Date)
  • Jossey-Bass

    (Publisher)

  One solution (7, 7, 1) was just discussed. But even this is not quite enough. You see, (2, 5, 8) would be another solution to the equa- tion, but you couldn’t form a triangle with those lengths. With 8 as one side, the two sides of 2 and 5 (whose sum is 7) wouldn’t “meet” or connect. So we need to add one more fact to our search, called the trian- gle inequality property. This says that in any triangle, the sum of the lengths of any two sides must exceed the length of the third side. In our good example, we have 7 + 1 > 7; but in our bad example, we have 2 + 5 < 8. With a little patience, we can systematically form a list of solutions: 1. 1-7-7 2. 2-6-7 3. 3-6-6 4. 3-5-7 5. 4-4-7 6. 4-5-6 7. 5-5-5 Triangles: Properties, Congruence, and Similarity 57 2 2 Note how we let a equal the smallest side and kept it constant as long as possible while looking for the lengths of b and c. This is just to get organized. If you are concerned about obtaining all the solu- tions, it helps to have a systematic way of searching for them, such as letting a be the length of the smallest side. —Dr. Math, The Math Forum Centers of Triangles When you talk about the center of a circle, there is only one possible point. If you’re talking about centers of other objects, it’s often a bit more complicated. How do you find the center of your backyard, for example? It certainly depends on what shape the yard is! In this sec- tion, we’ll look at the different types of centers of triangles. Hi, Quentin, To understand what facts we are given and what we need to prove, let’s review some definitions. The altitude, median, and angle bisec- tor of a triangle are all line segments that join one vertex of a trian- 58 Dr. Math Presents More Geometry Dear Dr. Math, I don’t understand the difference between angle bisectors, medians, and altitudes. Here’s a problem that I have to prove: In an isosceles triangle, the altitude is a median and an angle bisector.

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