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In geometry, an equilateral triangle is a triangle in which all three sides have the same length.

Properties

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An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides.[1] Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base.[2]

The follow-up definition above may result in more precise properties. For example, since the perimeter of an isosceles triangle is the sum of its two legs and its base,[3] the equilateral triangle is formulated as three times its side. The cevian of an equilateral triangle are all equal in length, resulting in the median and angle bisector being equal in length, considering those lines as their altitude depending on the base's choice.[4] When the equilateral triangle is flipped around its altitude and rotated around its center for every one-third of a full angle, its appearance is unchangeable. This leads that the equilateral triangle has the symmetry of a dihedral group   of order six.[5] The internal angle of an equilateral triangle are equal, 60°.[4] The following describes others.

Area

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The area of an equilateral triangle is   The formula may be derived from the formula of an isosceles triangle by Pythagoras theorem: the altitude   of a triangle is the square root of the difference of two squares of a side and half of a base.[3] Since the base and the legs are equal, the height is[citation needed]   In general, the area of a triangle is the half product of base and height. The formula of the area of an equilateral triangle can be obtained by substituting the altitude formula. Another way to prove the area of an equilateral triangle is by using the trigonometric function. The area of a triangle is formulated as the half product of base and height and the sine of an angle. Because all of the angles of an equilateral triangle are 60°, the formula is as desired.[citation needed]

Relationship with circles

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The radius of the circumscribed circle is:   and the radius of the inscribed circle is half of the circumradius:  

The theorem of Euler states that the distance   between circumradius and inradius is formulated as  . The aftermath results in a triangle inequality stating that the equilateral triangle has the smallest ratio of the circumradius   to the inradius   of any triangle. That is:[6]  

Pompeiu's theorem states that, if   is an arbitrary point in the plane of an equilateral triangle   but not on its circumcircle, then there exists a triangle with sides of lengths  ,  , and  . That is,  ,  , and   satisfy the triangle inequality that the sum of any two of them is greater than the third. If   is on the circumcircle then the sum of the two smaller ones equals the longest and the triangle has degenerated into a line, this case is known as Van Schooten's theorem.

A packing problem asks the objective of   circles packing into the smallest possible equilateral triangle. The optimal solutions show   that can be packed into the equilateral triangle, but the open conjectures expand to  .

Other mathematical properties

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Visual proof of Viviani's theorem

Morley's trisector theorem states that, in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.

Napoleon's theorem states that, if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those equilateral triangles themselves form an equilateral triangle.

A version of the isoperimetric inequality for triangles states that the triangle of greatest area among all those with a given perimeter is equilateral.[7]

Viviani's theorem states that, for any interior point   in an equilateral triangle with distances  ,  , and   from the sides and altitude  ,   independent of the location of  .[8]

Construction

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Construction of equilateral triangle with compass and straightedge

The equilateral triangle can be constructed in different ways by using circles. The first proposition in the Elements first book by Euclid. Start by drawing a circle with a certain radius, placing the point of the compass on the circle, and drawing another circle with the same radius; the two circles will intersect in two points. An equilateral triangle can be constructed by taking the two centers of the circles and either of the points of intersection.[9]

An alternative way to construct an equilateral triangle is by using Fermat prime. A Fermat prime is a prime number of the form   wherein   denotes the non-negative integer, and there are five known Fermat primes: 3, 5, 17, 257, 65537. A regular polygon is constructible by compass and straightedge if and only if the odd prime factors of its number of sides are distinct Fermat primes.[10] To do so geometrically, draw a straight line and place the point of the compass on one end of the line, then swing an arc from that point to the other point of the line segment; repeat with the other side of the line, which connects the point where the two arcs intersect with each end of the line segment in the aftermath.

Appearances

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Tillings, polyhedra, and polytopes

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The equilateral triangle tiling fills the plane
 
Regular octahedron is a deltahedron as well as the family of antiprism

Notably, the equilateral triangle tiles is a two-dimensional space with six triangles meeting at a vertex, whose dual tessellation is the hexagonal tiling. Truncated hexagonal tiling, rhombitrihexagonal tiling, trihexagonal tiling, snub square tiling, and snub hexagonal tiling are all semi-regular tessellations constructed with equilateral triangles.[11]

Equilateral triangles may also form a polyhedron in three dimensions. Three of five polyhedrons of Platonic solids are regular tetrahedron, regular octahedron, and regular icosahedron. Five of the Johnson solids are triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, and gyroelongated square bipyramid. All of these eight convex polyhedrons have the equilateral triangle as their faces, known as the deltahedron.[12] More generally, all Johnson solids have equilateral triangles, though there are some other regular polygons as their faces.[13] Antiprism is another family of polyhedra where all the faces other than the bases mostly consist of alternating triangles. When the antiprism is uniform, its bases are regular and all triangular faces are equilateral.[14]

As a generalization, the equilateral triangle belongs to the infinite family of  -simplexes, with  .[15]

Applications

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Equilateral triangle usage as a yield sign

Equilateral triangles have frequently appeared in man-made constructions and popular cultures. In architecture, an example can be seen in the cross-section of the Gateway Arch and the surface of a Vegreville egg.[16][17] The faces of Giza Pyramid may be seen as equilateral triangles, yet the resulting accurately shows they are most likely isosceles triangles instead.[18] In heraldic and flags, its applications include the flag of Nicaragua and the flag of the Philippines.[19][20] It is a shape of a variety of road signs, including the yield sign.[21]

The equilateral triangle occurred in the study of stereochemistry. It can be described as the molecular geometry in which one atom in the center connects three other atoms in a plane, known as the trigonal planar molecular geometry.

In the Thomson problem, concerning the minimum-energy configuration of   charged particles on a sphere, and for the Tammes problem of constructing a spherical code maximizing the smallest distance among the points, the minimum solution known for   places the points at the vertices of an equilateral triangle, inscribed in a sphere. This configuration is proven optimal for the Tammes problem, but a rigorous solution to this instance of the Thomson problem is unknown.[22]

References

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  1. ^ Stahl, Saul (2003). Geometry from Euclid to Knots. Prentice-Hall. ISBN 0-13-032927-4.
  2. ^ Lardner, Dionysius (1840). A Treatise on Geometry and Its Application in the Arts. London: The Cabinet Cyclopædia. p. 46.
  3. ^ a b Harris, John W.; Stocker, Horst (1998). Handbook of mathematics and computational science. New York: Springer-Verlag. p. 78. doi:10.1007/978-1-4612-5317-4. ISBN 0-387-94746-9. MR 1621531.
  4. ^ a b Owen, Byer; Felix, Lazebnik; Deirdre, Smeltzer (2010). Methods for Euclidean Geometry. Classroom Resource Materials. Vol. 37. Washington, D.C.: Mathematical Association of America. pp. 36, 39. doi:10.5860/choice.48-3331. ISBN 9780883857632. OCLC 501976971. S2CID 118179744.
  5. ^ Carstensen, Celine; Fine, Celine; Rosenberger, Gerhard (2011). Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography. De Gruyter. p. 156. ISBN 978-3-11-025009-1.
  6. ^ Svrtan, Dragutin; Veljan, Darko (2012). "Non-Euclidean versions of some classical triangle inequalities". Forum Geometricorum. 12: 197–209.
  7. ^ Cite error: The named reference Chakerian was invoked but never defined (see the help page).
  8. ^ Posamentier, Alfred S.; Salkind, Charles T. (1996). Challenging Problems in Geometry. Dover Publ.
  9. ^ Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press. p. 62. ISBN 978-0-521-55432-9.
  10. ^ Křížek, Michal; Luca, Florian; Somer, Lawrence (2001). 17 Lectures on Fermat Numbers: From Number Theory to Geometry. CMS Books in Mathematics. Vol. 9. New York: Springer-Verlag. pp. 1–2. doi:10.1007/978-0-387-21850-2. ISBN 978-0-387-95332-8. MR 1866957.
  11. ^ Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 231–234. doi:10.2307/2689529. JSTOR 2689529. MR 1567647. S2CID 123776612. Zbl 0385.51006.
  12. ^ Trigg, Charles W. (1978). "An infinite class of deltahedra". Mathematics Magazine. 51 (1): 55–57. doi:10.1080/0025570X.1978.11976675. JSTOR 2689647. MR 1572246.
  13. ^ Berman, Martin (1971). "Regular-faced convex polyhedra". Journal of the Franklin Institute. 291 (5): 329–352. doi:10.1016/0016-0032(71)90071-8. MR 0290245.
  14. ^ Horiyama, Takayama; Itoh, Jin-ichi; Katoh, Naoi; Kobayashi, Yuki; Nara, Chie (14–16 September 2015). "Continuous Folding of Regular Dodecahedra". In Akiyama, Jin; Ito, Hiro; Sakai, Toshinori; Uno, Yushi (eds.). Discrete and Computational Geometry and Graphs. Japanese Conference on Discrete and Computational Geometry and Graphs. Kyoto. p. 124. doi:10.1007/978-3-319-48532-4. ISBN 978-3-319-48532-4. ISSN 1611-3349.{{cite conference}}: CS1 maint: date format (link)
  15. ^ Coxeter, = H. S. M. Coxeter (1948). Regular Polytopes (1 ed.). London: Methuen & Co. LTD. pp. 120–121. OCLC 4766401. Zbl 0031.06502.{{cite book}}: CS1 maint: extra punctuation (link)
  16. ^ Pelkonen, Eeva-Liisa; Albrecht, Donald, eds. (2006). Eero Saarinen: Shaping the Future. Yale University Press. pp. 160, 224, 226. ISBN 978-0972488129.
  17. ^ Alsina, Claudi; Nelsen, Roger B. (2015). A Mathematical Space Odyssey: Solid Geometry in the 21st Century. Vol. 50. Mathematical Association of America. p. 22. ISBN 978-1-61444-216-5.
  18. ^ Herz-Fischler, Roger (2000). The Shape of the Great Pyramid. Wilfrid Laurier University Press. p. 46. ISBN 0-88920-324-5.
  19. ^ White, Steven F.; Calderón, Esthela (2008). Culture and Customs of Nicaragua. Greenwood Press. p. 3. ISBN 978-0313339943.
  20. ^ Guillermo, Artemio R. (2012). Historical Dictionary of the Philippines. Scarecrow Press. p. 161. ISBN 978-0810872462.
  21. ^ Riley, Michael W.; Cochran, David J.; Ballard, John L. (December 1982). "An Investigation of Preferred Shapes for Warning Labels". Human Factors: The Journal of the Human Factors and Ergonomics Society. 24 (6): 737–742. doi:10.1177/001872088202400610. S2CID 109362577.
  22. ^ Whyte, L. L. (1952). "Unique arrangements of points on a sphere". The American Mathematical Monthly. 59 (9): 606–611. doi:10.1080/00029890.1952.11988207. JSTOR 2306764. MR 0050303.

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