User:Dedhert.Jr/sandbox/7: Difference between revisions

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.{{r|stahl}} Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base.{{r|lardner}}

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 base, the equilateral triangle is formulated as three times its side.{{r|harris-stocker|cerin}} 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.{{r|ofd}} 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]] <math> D_3 </math> of order six.{{r|cfr}} The [[internal angle]] of an equilateral triangle are equal, 60&deg;.{{r|ofd}} The following describes others.

The formula may be derived from the formula of an isosceles triangle by [[Pythagoras theorem]]: the altitude <math> h </math> of a triangle is [[Isosceles triangle#Height|the square root of the difference of two squares of a side and half of a base]].{{r|harris-stocker}} Since the base and the legs are equal, the height is:{{r|mp}}

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.{{r|mp}} 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&deg;, the formula is as desired.{{cn}}

A version of the [[isoperimetric inequality#Isoperimetric inequality for triangles|isoperimetric inequality]] for triangles states that the triangle of greatest [[area]] among all those with a given [[perimeter]] is equilateral.{{r|chakerian}}

The [[Euler's theorem in geometry|theorem of Euler]] states that the distance <math> t </math> between circumradius and inradius is formulated as <math> t^2 = R(R - 2r) </math>. The aftermath results in a triangle inequality stating that the equilateral triangle has the smallest ratio of the circumradius <math>R</math> to the inradius <math>r</math> of any triangle. That is:{{r|sv}}

[[Pompeiu's theorem]] states that, if <math>P</math> is an arbitrary point in the plane of an equilateral triangle <math>ABC</math> but not on its [[circumcircle]], then there exists a triangle with sides of lengths <math>PA</math>, <math>PB</math>, and <math>PC</math>. That is, <math>PA</math>, <math>PB</math>, and <math>PC</math> satisfy the [[triangle inequality]] that the sum of any two of them is greater than the third. If <math>P</math> 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]].

The equilateral triangle can be constructed in different ways by using circles. The first proposition in the [[Euclid's Elements|''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.{{r|cromwell}}

wherein <math> k </math> 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.{{r|kls}} 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.

=== Tillings, polyhedra, and polytopes ===

Notably, the equilateral triangle [[Euclidean tilings by convex regular polygons#Regular tilings|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 [[Euclidean tilings by convex regular polygons#Archimedean, uniform or semiregular tilings|semi-regular tessellations]] constructed with equilateral triangles.{{r|gs}}

Equilateral triangles may also form a polyhedron in three dimensions. Three of five polyhedrons of [[Platonic solid|Platonic solids]] are [[regular tetrahedron]], [[regular octahedron]], and [[regular icosahedron]]. Five of the [[Johnson solid|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]].{{r|trigg}} More generally, all [[Johnson solid]]s have equilateral triangles, though there are some other [[regular polygon]]s as their faces.{{r|berman}} [[Antiprism]] is another family of polyhedra where all the faces other than the bases mostly consist of alternating triangles. When the antiprism is [[Uniform polyhedron|uniform]], its bases are regular and all triangular faces are equilateral.{{r|antiprism}}

As a generalization, the equilateral triangle belongs to the infinite family of <math>n</math>-[[simplex (geometry)|simplexes]], with <math>n = 2</math>.{{r|coxeter}}

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]].{{r|gateway|an}} The faces of [[Giza Pyramid]] may be seen as equilateral triangles, yet the resulting accurately shows they are most likely isosceles triangles instead.{{r|herz}} In heraldic and flags, its applications include the [[flag of Nicaragua]] and the [[flag of the Philippines]].{{r|wc|guillermo}} It is a shape of a variety of [[traffic sign|road signs]], including the [[yield sign]].{{r|rcb}}

In the [[Thomson problem]], concerning the minimum-energy configuration of <math>n</math> 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 <math>n=3</math> places the points at the vertices of an equilateral triangle, [[Circumscribed sphere|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.{{r|whyte}}

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

| editor1-first = Eeva-Liisa | editor1-last = Pelkonen

| editor2-first = Donald | editor2-last = Albrecht

| publisher = Yale University Press

| url = https://archive.org/details/eerosaarinenshap0000saar/page/160

}}</ref>

| first1 = Branko | last1 = Grünbaum | author-link = Branko Grünbaum

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

}}</ref>

| volume = 59