Isosceles triangle
Isosceles triangle | |
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cyclic | |
Dual polygon | Self-dual |
In geometry, an isosceles triangle (/aɪˈsɒsəliːz/) is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the
The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings.
The two equal sides are called the legs and the third side is called the base of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated by simple formulas from the lengths of the legs and base. Every isosceles triangle has an axis of symmetry along the
Terminology, classification, and examples
In an isosceles triangle that has exactly two equal sides, the equal sides are called
Whether an isosceles triangle is
As well as the
Formulas
Height
For any isosceles triangle, the following six line segments coincide:
- the altitude, a line segment from the apex perpendicular to the base,[14]
- the angle bisector from the apex to the base,[14]
- the median from the apex to the midpoint of the base,[14]
- the perpendicular bisector of the base within the triangle,[14]
- the segment within the triangle of the unique axis of symmetry of the triangle, and[14]
- the segment within the triangle of the Euler line of the triangle, except when the triangle is equilateral.[15]
Their common length is the height of the triangle. If the triangle has equal sides of length and base of length , the general triangle formulas for the lengths of these segments all simplify to[16]
This formula can also be derived from the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles.[17]
The Euler line of any triangle goes through the triangle's
Area
The area of an isosceles triangle can be derived from the formula for its height, and from the general formula for the area of a triangle as half the product of base and height:[16]
The same area formula can also be derived from
If the apex angle and leg lengths of an isosceles triangle are known, then the area of that triangle is:[20]
This is a special case of the general formula for the area of a triangle as half the product of two sides times the sine of the included angle.[21]
Perimeter
The perimeter of an isosceles triangle with equal sides and base is just[16]
As in any triangle, the area and perimeter are related by the isoperimetric inequality[22]
This is a strict inequality for isosceles triangles with sides unequal to the base, and becomes an equality for the equilateral triangle. The area, perimeter, and base can also be related to each other by the equation[23]
If the base and perimeter are fixed, then this formula determines the area of the resulting isosceles triangle, which is the maximum possible among all triangles with the same base and perimeter.[24] On the other hand, if the area and perimeter are fixed, this formula can be used to recover the base length, but not uniquely: there are in general two distinct isosceles triangles with given area and perimeter . When the isoperimetric inequality becomes an equality, there is only one such triangle, which is equilateral.[25]
Angle bisector length
If the two equal sides have length and the other side has length , then the internal
as well as
and conversely, if the latter condition holds, an isosceles triangle parametrized by and exists.[27]
The Steiner–Lehmus theorem states that every triangle with two angle bisectors of equal lengths is isosceles. It was formulated in 1840 by C. L. Lehmus. Its other namesake, Jakob Steiner, was one of the first to provide a solution.[28] Although originally formulated only for internal angle bisectors, it works for many (but not all) cases when, instead, two external angle bisectors are equal. The 30-30-120 isosceles triangle makes a
Radii
The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles.[30] The radius of the
The center of the circle lies on the symmetry axis of the triangle, this distance above the base. An isosceles triangle has the largest possible inscribed circle among the triangles with the same base and apex angle, as well as also having the largest area and perimeter among the same class of triangles.[31]
The radius of the circumscribed circle is:[16]
The center of the circle lies on the symmetry axis of the triangle, this distance below the apex.
Inscribed square
For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base square.[10] A much older theorem, preserved in the works of Hero of Alexandria, states that, for an isosceles triangle with base and height , the side length of the inscribed square on the base of the triangle is[32]
Isosceles subdivision of other shapes
For any integer , any triangle can be partitioned into isosceles triangles.[33] In a right triangle, the median from the hypotenuse (that is, the line segment from the midpoint of the hypotenuse to the right-angled vertex) divides the right triangle into two isosceles triangles. This is because the midpoint of the hypotenuse is the center of the circumcircle of the right triangle, and each of the two triangles created by the partition has two equal radii as two of its sides.[34] Similarly, an
Generalizing the partition of an acute triangle, any
Either diagonal of a rhombus divides it into two congruent isosceles triangles. Similarly, one of the two diagonals of a kite divides it into two isosceles triangles, which are not congruent except when the kite is a rhombus.[37]
Applications
In architecture and design
Isosceles triangles commonly appear in architecture as the shapes of gables and pediments. In ancient Greek architecture and its later imitations, the obtuse isosceles triangle was used; in Gothic architecture this was replaced by the acute isosceles triangle.[8]
In the
Warren truss structures, such as bridges, are commonly arranged in isosceles triangles, although sometimes vertical beams are also included for additional strength.[40] Surfaces tessellated by obtuse isosceles triangles can be used to form deployable structures that have two stable states: an unfolded state in which the surface expands to a cylindrical column, and a folded state in which it folds into a more compact prism shape that can be more easily transported.[41] The same tessellation pattern forms the basis of Yoshimura buckling, a pattern formed when cylindrical surfaces are axially compressed,[42] and of the Schwarz lantern, an example used in mathematics to show that the area of a smooth surface cannot always be accurately approximated by polyhedra converging to the surface.[43]
In
They also have been used in designs with religious or mystic significance, for instance in the Sri Yantra of Hindu meditational practice.[47]
In other areas of mathematics
If a
In
History and fallacies
Long before isosceles triangles were studied by the ancient Greek mathematicians, the practitioners of Ancient Egyptian mathematics and Babylonian mathematics knew how to calculate their area. Problems of this type are included in the Moscow Mathematical Papyrus and Rhind Mathematical Papyrus.[50]
The theorem that the base angles of an isosceles triangle are equal appears as Proposition I.5 in Euclid.[51] This result has been called the pons asinorum (the bridge of asses) or the isosceles triangle theorem. Rival explanations for this name include the theory that it is because the diagram used by Euclid in his demonstration of the result resembles a bridge, or because this is the first difficult result in Euclid, and acts to separate those who can understand Euclid's geometry from those who cannot.[52]
A well-known fallacy is the false proof of the statement that all triangles are isosceles, first published by W. W. Rouse Ball in 1892,[53] and later republished in Lewis Carroll's posthumous Lewis Carroll Picture Book.[54] The fallacy is rooted in Euclid's lack of recognition of the concept of betweenness and the resulting ambiguity of inside versus outside of figures.[55]
Notes
- ^ Heath (1956), p. 187, Definition 20.
- ^ Stahl (2003), p. 37.
- ^ Usiskin & Griffin (2008), p. 4.
- ^ Usiskin & Griffin (2008), p. 41.
- ^ Ionin (2009).
- ^ Jacobs (1974), p. 144.
- ^ a b Gottschau, Haverkort & Matzke (2018).
- ^ a b c d Lardner (1840), p. 46.
- ^ Barnes (2012).
- ^ a b Conway & Guy (1996).
- ^ Loeb (1992).
- ^ Langley (1922).
- ^ Montroll (2009).
- ^ a b c d e Hadamard (2008), p. 23.
- ^ a b Guinand (1984).
- ^ a b c d e Harris & Stöcker (1998), p. 78.
- ^ Salvadori & Wright (1998).
- ^ Hadamard (2008), Exercise 5, p. 29.
- ^ Kahan (2014).
- ^ Young (2011), p. 298.
- ^ Young (2011), p. 398.
- ^ Alsina & Nelsen (2009), p. 71.
- ^ Baloglou & Helfgott (2008), Equation (1).
- ^ Wickelgren (2012).
- ^ Baloglou & Helfgott (2008), Theorem 2.
- ^ Arslanagić.
- ^ Oxman (2005).
- ^ Gilbert & MacDonnell (1963).
- ^ Conway & Ryba (2014).
- ^ a b Harris & Stöcker (1998), p. 75.
- ^ Alsina & Nelsen (2009), p. 67.
- ^ Gandz (1940).
- ^ Lord (1982). See also Hadamard (2008, Exercise 340, p. 270).
- ^ Posamentier & Lehmann (2012), p. 24.
- ^ Bezdek & Bisztriczky (2015).
- ^ Robbins (1995).
- ^ Usiskin & Griffin (2008), p. 51.
- ^ Lavedan (1947).
- ^ Padovan (2002).
- ^ Ketchum (1920).
- ^ Pellegrino (2002).
- ^ Yoshimura (1955).
- ^ Schwarz (1890).
- ^ Washburn (1984).
- ^ Jakway (1922).
- ^ Smith (2014).
- ^ Bolton, Nicol & Macleod (1977).
- ^ Bardell (2016).
- ^ Diacu & Holmes (1999).
- Vasily Vasilievich Struve championed the view that they used the correct formula, half the product of the base and height (Clagett 1989). This question rests on the translation of one of the words in the Rhind papyrus, and with this word translated as height (or more precisely as the ratio of height to base) the formula is correct (Gunn & Peet 1929, pp. 173–174).
- ^ Heath (1956), p. 251.
- ^ Venema (2006), p. 89.
- ^ Ball & Coxeter (1987).
- ^ Carroll (1899). See also Wilson (2008).
- ^ Specht et al. (2015).
References
- Alsina, Claudi; Nelsen, Roger B. (2009), When less is more: Visualizing basic inequalities, The Dolciani Mathematical Expositions, vol. 36, Mathematical Association of America, Washington, DC, MR 2498836
- Arslanagić, Šefket, "Problem η44", Inequalities proposed in Crux Mathematicorum (PDF), p. 151
- ISBN 0-486-25357-0
- Baloglou, George; Helfgott, Michel (2008), "Angles, area, and perimeter caught in a cubic" (PDF), MR 2373294
- Bardell, Nicholas S. (2016), "Cubic polynomials with real or complex coefficients: The full picture" (PDF), Australian Senior Mathematics Journal, 30 (2): 5–26
- Barnes, John (2012), Gems of Geometry (2nd, illustrated ed.), Springer, p. 27, ISBN 9783642309649
- Bezdek, András; Bisztriczky, Ted (2015), "Finding equal-diameter triangulations in polygons", Beiträge zur Algebra und Geometrie, 56 (2): 541–549, S2CID 123507725
- Bolton, Nicholas J; Nicol, D.; Macleod, G. (March 1977), "The geometry of the Śrī-yantra", Religion, 7 (1): 66–85,
- Carroll, Lewis (1899), Collingwood, Stuart Dodgson (ed.), The Lewis Carroll Picture Book, London: T. Fisher Unwin, pp. 264–266
- ISBN 9780871692320
- Conway, J.H.; Guy, R.K. (1996), "Calabi's Triangle", The Book of Numbers, New York: Springer-Verlag, p. 206
- S2CID 124753764
- Diacu, Florin; Holmes, Philip (1999), Celestial Encounters: The Origins of Chaos and Stability, Princeton Science Library, Princeton University Press, p. 122, ISBN 9780691005454
- Gandz, Solomon (1940), "Studies in Babylonian mathematics. III. Isoperimetric problems and the origin of the quadratic equations", Isis, 32: 101–115 (1947), S2CID 120267556. See in particular p. 111.
- Gilbert, G.; MacDonnell, D. (1963), "The Steiner–Lehmus Theorem", Classroom Notes, MR 1531983
- Gottschau, Marinus; Haverkort, Herman; Matzke, Kilian (2018), "Reptilings and space-filling curves for acute triangles", S2CID 14477196
- Guinand, Andrew P. (1984), "Euler lines, tritangent centers, and their triangles", American Mathematical Monthly, 91 (5): 290–300, MR 0740243
- Gunn, Battiscombe; Peet, T. Eric (May 1929), "Four geometrical problems from the Moscow Mathematical Papyrus", The Journal of Egyptian Archaeology, 15 (1): 167–185, S2CID 192278129
- ISBN 9780821843673
- Harris, John W.; MR 1621531
- ISBN 0-486-60088-2
- Høyrup, Jens (2008), "Geometry in Mesopotamia and Egypt", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures, Springer Netherlands, pp. 1019–1023,
- Ionin, Yury J. (2009), "Isosceles sets", MR 2577309
- Jacobs, Harold R. (1974), Geometry, W. H. Freeman and Co., ISBN 0-7167-0456-0
- Jakway, Bernard C. (1922), The Principles of Interior Decoration, Macmillan, p. 48
- Kahan, W. (September 4, 2014), "Miscalculating Area and Angles of a Needle-like Triangle" (PDF), Lecture Notes for Introductory Numerical Analysis Classes, University of California, Berkeley
- Ketchum, Milo Smith (1920), The Design of Highway Bridges of Steel, Timber and Concrete, New York: McGraw-Hill, p. 107
- Langley, E. M. (1922), "Problem 644", The Mathematical Gazette, 11: 173
- Lardner, Dionysius (1840), A Treatise on Geometry and Its Application in the Arts, The Cabinet Cyclopædia, London
- Lavedan, Pierre (1947), French Architecture, Penguin Books, p. 44
- ISBN 0-8176-3620-X
- Lord, N. J. (June 1982), "66.16 Isosceles subdivisions of triangles", S2CID 125411311
- ISBN 9781439871065
- Oxman, Victor (2005), "On the existence of triangles with given lengths of one side, the opposite and one adjacent angle bisectors" (PDF), MR 2141652
- ISBN 9780415259620
- Pellegrino, S. (2002), Deployable Structures, CISM International Centre for Mechanical Sciences, vol. 412, Springer, pp. 99–100, ISBN 9783211836859
- MR 2963520
- Robbins, David P. (1995), "Areas of polygons inscribed in a circle", American Mathematical Monthly, 102 (6): 523–530, MR 1336638
- Salvadori, Mario; Wright, Joseph P. (1998), Math Games for Middle School: Challenges and Skill-Builders for Students at Every Level, Chicago Review Press, pp. 70–71, ISBN 9781569767276
- Schwarz, H. A. (1890), Gesammelte Mathematische Abhandlungen von H. A. Schwarz, Verlag von Julius Springer, pp. 309–311
- Smith, Whitney (June 26, 2014), "Flag of Saint Lucia", Encyclopædia Britannica, retrieved 2018-09-12
- Specht, Edward John; Jones, Harold Trainer; Calkins, Keith G.; Rhoads, Donald H. (2015), Euclidean geometry and its subgeometries, Springer, Cham, p. 64, MR 3445044
- Stahl, Saul (2003), Geometry from Euclid to Knots, Prentice-Hall, ISBN 0-13-032927-4
- ISBN 9781607526001
- Venema, Gerard A. (2006), Foundations of Geometry, Prentice-Hall, ISBN 0-13-143700-3
- Washburn, Dorothy K. (July 1984), "A study of the red on cream and cream on red designs on Early Neolithic ceramics from Nea Nikomedeia", American Journal of Archaeology, 88 (3): 305–324, S2CID 191374019
- Wickelgren, Wayne A. (2012), How to Solve Mathematical Problems, Dover Books on Mathematics, Courier Corporation, pp. 222–224, ISBN 9780486152684.
- MR 2455534
- Yoshimura, Yoshimaru (July 1955), On the mechanism of buckling of a circular cylindrical shell under axial compression, Technical Memorandum 1390, National Advisory Committee for Aeronautics
- ISBN 9780470648025