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| {{Infobox Polygon
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| | name = Equilateral triangle
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| | image = Triangle.Equilateral.svg
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| | type = [[Regular polygon]]
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| | edges = 3
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| | schläfli = {3}
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| | coxeter = {{CDD|node_1|3|node}}
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| | symmetry = [[dihedral symmetry|D<sub>3</sub>]]
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| | area = <math>\tfrac{\sqrt{3}}{4} a^2</math>
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| | angle = 60°}}
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| In [[geometry]], an '''equilateral triangle''' is a [[triangle]] in which all three sides are equal. In traditional or [[Euclidean geometry]], equilateral triangles are also [[Equiangular polygon|equiangular]]; that is, all three internal angles are also congruent to each other and are each 60°. They are [[regular polygon]]s, and can therefore also be referred to as regular triangles.
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| ==Principal properties==
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| [[File:Equilateral-triangle-heights.svg|300px|thumb|Equilateral triangle]]
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| <!--Please do not change the formulas they are correct; check your work if you do-->
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| Assuming the lengths of the sides of the equilateral triangle are ''a'', we can determine using the [[Pythagorean theorem]] that:
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| * The area is <math>A=\frac{\sqrt{3}}{4} a^2</math>
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| * The perimeter is <math>p=3a\,\!</math>
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| * The radius of the [[circumscribed circle]] is <math>R=\frac{\sqrt{3}}{3} a</math>
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| * The radius of the [[inscribed circle]] is <math>r=\frac{\sqrt{3}}{6} a</math>
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| * The geometric center of the triangle is the center of the circumscribed and inscribed circles
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| * And the [[Altitude (triangle)|altitude]] (height) from any side is <math>h=\frac{\sqrt{3}}{2} a</math>.
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| <br />
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| Many of these quantities have simple relationships to the altitude ("h") of each vertex from the opposite side:
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| * The area is <math>A=\frac{h^2}{2}</math>
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| * The height of the center from each side is <math>A=\frac{h}{3} </math>
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| * The radius of the circle circumscribing the three vertices is <math>R=\frac{2h}{3} </math>
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| * The radius of the inscribed circle is <math>r=\frac{h}{3}</math>
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| <br />
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| In an equilateral triangle, the altitudes, the angle bisectors, the perpendicular bisectors and the medians to each side coincide.
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| ==Derivation of area formula==
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| ===Using Pythagorean theorem===
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| The area of a triangle is half the base times the height.
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| <math>A = \frac{1}{2} bh</math>
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| The height of an equilateral triangle can be found using [[Pythagorean theorem]].
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| [[Image:30-60-90.svg|thumb|150px|An equilateral triangle with a side of 2 has a height of {{sqrt|3}} and the sine of 60° is {{sqrt|3}}/2.]]
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| The legs of the right triangle are half of the base and the height, while the hypotenuse is the side or base of the equilateral triangle.
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| <math>(\frac{b}{2})^2 + h^2 = b^2</math>
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| The formula for the height can be derived by rearranging the equation and finding the square root.
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| <math>h = \sqrt{b^2 - (\frac{b}{2})^2} = \sqrt{b^2 - \frac{b^2}{4}} = \sqrt{\frac{3}{4}b^2} = \frac{\sqrt{3}}{2}b</math>
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| By substituting <math>h = {\tfrac{\sqrt{3}}{2}b}</math>, the formula for the area of an equilateral triangle can be derived.
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| <math>A = \frac{1}{2}b \times \frac{\sqrt{3}}{2}b = \frac{\sqrt{3}}{4}b^2</math>
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| ===Using trigonometry===
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| Using trigonometry, the area of a triangle with any two sides a and b, and an angle C between them is:
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| <math>A = \frac{1}{2} ab \sin C</math>
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| Each angle of an equilateral triangle is 60°.
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| <math>A = \frac{1}{2} ab \sin 60^\circ</math>
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| The sine of 60° is <math>\tfrac{\sqrt{3}}{2}</math>.
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| <math>A = \frac{1}{2} ab \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4}ab</math>
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| All sides of an equilateral triangle are equal, therefore a = b.
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| <math>A = \frac{\sqrt{3}}{4}a^2</math>
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| ==Characterizations==
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| A triangle ''ABC'' that has the sides ''a'', ''b'', ''c'', [[semiperimeter]] ''s'', [[area]] ''T'', [[Incircle and excircles of a triangle#Relation to area of the triangle|exradii]] ''r<sub>a</sub>'', ''r<sub>b</sub>'', ''r<sub>c</sub>'' (tangent to ''a'', ''b'', ''c'' respectively), and where ''R'' and ''r'' are the radii of the [[Circumscribed circle#Circumscribed circles of triangles|circumcircle]] and [[Incircle and excircles of a triangle|incircle]] respectively, is equilateral [[if and only if]] any one of the statements in the following eight categories is true. These are also properties of an equilateral triangle.
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| ===Sides===
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| *<math>\displaystyle a^2+b^2+c^2=ab+bc+ca.</math><ref name=Andreescu/>
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| *<math>\displaystyle abc=(a+b-c)(a-b+c)(-a+b+c)\quad\text{(Lehmus)}</math><ref name=Cosmin/>
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| *<math>\displaystyle \frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{\sqrt{25Rr-2r^2}}{4Rr}.</math><ref>M. Bencze, Hui-Hua Wu and Shan-He Wu, "An equivalent form of fundamental triangle inequality and its applications", ''Research Group in Mathematical Inequalities and Applications'', Volume 11, Issue 1, 2008, [http://rgmia.org/papers/v11n1/equivalent.pdf]</ref>
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| ===Semiperimeter===
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| *<math>\displaystyle s=2R+(3\sqrt{3}-4)r\quad\text{(Blundon)}</math><ref>G. Dospinescu, M. Lascu, C. Pohoata & M. Letiva, "An elementary proof of Blundon's inequality", ''Journal of inequalities in pure and applied mathematics'', vol. 9, iss. 4, 2008, [http://www.emis.de/journals/JIPAM/images/220_08_JIPAM/220_08_www.pdf]</ref>
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| *<math>\displaystyle s^2=3r^2+12Rr.</math><ref>Blundon, W. J., "On Certain Polynomials Associated with the Triangle", ''Mathematics Magazine'', Vol. 36, No. 4 (Sep., 1963), pp. 247-248.</ref>
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| *<math>\displaystyle s^2=3\sqrt{3}T.</math><ref name=Alsina>Alsina, Claudi & Nelsen, Roger B., ''When less is more. Visualizing basic inequalities'', Mathematical Association of America, 2009, pp. 71, 155.</ref>
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| *<math>\displaystyle s=3\sqrt{3}r</math>
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| *<math>\displaystyle s=\frac{3\sqrt{3}}{2}R</math>
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| ===Angles===
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| *<math>\displaystyle A=B=C=60^\circ</math>
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| *<math>\displaystyle \cos{A}+\cos{B}+\cos{C}=\frac{3}{2}</math>
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| *<math>\displaystyle \sin{\frac{A}{2}}\sin{\frac{B}{2}}\sin{\frac{C}{2}}=\frac{1}{8}.</math><ref name=Cosmin/>
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| ===Area===
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| *<math>\displaystyle T=\frac{a^2+b^2+c^2}{4\sqrt{3}}\quad\text{(Weizenbock)}</math><ref>Cam McLeman & Andrei Ismail, "Weizenbock's inequality", PlanetMath, [http://planetmath.org/encyclopedia/WeizenbocksInequality.html].</ref>
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| *<math>\displaystyle T=\frac{\sqrt{3}}{4}(abc)^{^{\frac{2}{3}}}.</math><ref name=Alsina/>
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| ===Circumradius, inradius and exradii===
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| *<math>\displaystyle R=2r\quad\text{(Chapple-Euler)}</math><ref name=Andreescu/>
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| *<math>\displaystyle 9R^2=a^2+b^2+c^2.</math><ref name=Andreescu/>
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| *<math>\displaystyle r=\frac{r_a+r_b+r_c}{9}.</math><ref name=Cosmin>Pohoata, Cosmin, "A new proof of Euler's inradius - circumrdius inequality", ''Gazeta Matematica Seria B'', no. 3, 2010, pp. 121-123, [http://rms.unibuc.ro/gazeta/gmb/2010/3/articol.pdf].</ref>
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| *<math>\displaystyle r_a=r_b=r_c.</math>
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| ===Equal cevians===
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| Three kinds of [[cevian]]s are equal for (and only for) equilateral triangles:<ref>Byer, Owen; Lazebnik, Felix and Smeltzer, Deirdre, ''Methods for Euclidean Geometry'', Mathematical Association of America, 2010, pp. 36, 39.</ref>
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| *The three [[altitude (triangle)|altitude]]s have equal lengths.
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| *The three [[median (geometry)|median]]s have equal lengths.
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| *The three [[Bisection#Angle bisector|angle bisector]]s have equal lengths.
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| ===Coincident triangle centers===
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| Every [[triangle center]] of an equilateral triangle coincides with its [[centroid]], and for some pairs of triangle centers, the fact that they coincide is enough to ensure that the triangle is equilateral. In particular, a triangle is equilateral if any two of the [[Circumscribed circle|circumcenter]], [[Incircle and excircles of a triangle|incenter]], centroid, or [[Altitude (triangle)|orthocenter]] coincide.<ref>Yiu, Paul, ''Notes on Euclidean Geometry'', 1998, p. 37, [http://www.math.fau.edu/Yiu/EuclideanGeomeryNotes.pdf]</ref> It is also equilateral if its circumcenter coincides with the [[Nagel point]], or if its incenter coincides with its [[Nine-point circle#Other interesting facts|nine-point center]].<ref name=Andreescu>Andreescu, Titu and Andrica, Dorian, "Complex Numbers from A to...Z", Birkhäuser, 2006, pp. 70, 113-115.</ref>
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| ===Six triangles formed by partitioning by the medians===
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| For any triangle, the three [[median (geometry)|medians]] partition the triangle into six smaller triangles.
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| *A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.<ref name=Cerin>[http://forumgeom.fau.edu/FG2004volume4/FG200413.pdf Cˇerin, Zvonko, "The vertex-midpoint-centroid triangles", ''Forum Geometricorum'' 4, 2004: pp. 97–109.]</ref>{{rp|Theorem 1}}
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| *A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.<ref name=Cerin/>{{rp|Corollary 7}}
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| ==Famous theorems==
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| [[Morley's trisector theorem]] states that, in any triangle, the three points of intersection of the adjacent [[Angle trisection|angle trisectors]] form an equilateral triangle.
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| [[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.
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| A version of the [[isoperimetric inequality]] for triangles states that the triangle of greatest [[area]] among all those with a given [[perimeter]] is equilateral.<ref name=Chakerian/>
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| [[Viviani's theorem]] states that, for any interior point ''P'' in an equilateral triangle, with distances ''d'', ''e'', and ''f'' from the sides, ''d'' + ''e'' + ''f'' = the altitude of the triangle, independent of the location of ''P''.<ref name=Posamentier>Posamentier, Alfred S., and Salkind, Charles T., ''Challenging Problems in Geometry'', Dover Publ., 1996.</ref>
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| [[Pompeiu's theorem]] states that, if ''P'' is an arbitrary point in an equilateral triangle ''ABC'', then there exists a triangle with sides of length ''PA'', ''PB'', and ''PC''.
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| ==Other properties==
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| By [[Euler's theorem in geometry|Euler's inequality]], the equilateral triangle has the smallest ratio ''R''/''r'' of the circumradius to the inradius of any triangle: specifically, ''R''/''r'' = 2.
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| The triangle of largest area of all those inscribed in a given circle is equilateral; and the triangle of smallest area of all those circumscribed around a given circle is equilateral.<ref>Dorrie, Heinrich, ''100 Great Problems of Elementary Mathematics'', Dover Publ., 1965: 379-380.</ref>
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| The ratio of the area of the incircle to the area of an equilateral triangle, <math>\frac{\pi}{3\sqrt{3}}</math>, is larger than that of any non-equilateral triangle.<ref>Minda, D., and Phelps, S., "Triangles, ellipses, and cubic polynomials", ''[[American Mathematical Monthly]]'' 115, October 2008, 679-689: Theroem 4.1.</ref>
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| The ratio of the area to the square of the perimeter of an equilateral triangle, <math>\frac{1}{12\sqrt{3}},</math> is larger than that for any other triangle.<ref name=Chakerian>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979: 147.</ref>
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| Given a point in the interior of an equilateral triangle, the ratio of the sum of its distances from the vertices to the sum of its distances from the sides equals 2 and is less than that of any other triangle.<ref>[http://forumgeom.fau.edu/FG2001volume1/FG200102.pdf Lee, Hojoo, "Another proof of the Erdős–Mordell Theorem", ''Forum Geometricorum'' 1, 2001: 7-8.]</ref> This is the [[Erdős–Mordell inequality]]; a stronger variant of it is [[Barrow's inequality]], which replaces the perpendicular distances to the sides with the distances to the points where the angle bisectors cross the sides.
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| For any point P in the plane, with distances ''p'', ''q'', and ''t'' from the vertices A, B, and C respectively,<ref name="De">De, Prithwijit, "Curious properties of the circumcircle and incircle of an equilateral triangle," ''Mathematical Spectrum'' 41(1), 2008-2009, 32-35.</ref>
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| :<math>\displaystyle 3(p^{4}+q^{4}+t^{4}+a^{4})=(p^{2}+q^{2}+t^{2}+a^{2})^{2}.</math>
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| For any point P on the inscribed circle of an equilateral triangle, with distances ''p'', ''q'', and ''t'' from the vertices,<ref name="De"/>
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| :<math>\displaystyle 4(p^{2}+q^{2}+t^{2})=5a^{2}</math>
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| and
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| :<math>\displaystyle 16(p^{4}+q^{4}+t^{4})=11a^{4}.</math>
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| For any point P on the minor arc BC of the circumcircle, with distances ''p'', ''q'', and ''t'' from A, B, and C respectively,<ref name="De"/>
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| :<math>\displaystyle p=q+t</math>
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| and
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| :<math>\displaystyle q^{2}+qt+t^{2}=a^{2} ;</math>
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| moreover, if point D on side BC divides PA into segments PD and DA with DA having length ''z'' and PD having length ''y'', then<ref name=Posamentier/>
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| :<math>z= \frac{t^{2}+tq+q^2}{t+q},</math>
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| which also equals <math>\tfrac{t^{3}-q^{3}}{t^{2}-q^{2}}</math> if ''t'' ≠ ''q''; and
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| :<math>\frac{1}{q}+\frac{1}{t}=\frac{1}{y}.</math>
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| An equilateral triangle is the most symmetrical triangle, having 3 lines of [[reflection symmetry|reflection]] and [[rotational symmetry]] of order 3 about its center. Its [[symmetry group]] is the [[dihedral group of order 6]] ''D''<sub>3</sub>.
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| Equilateral triangles are the only triangles whose [[Steiner inellipse]] is a circle (specifically, it is the incircle).
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| [[Image:Viervlak-frame.jpg|thumb|right|A regular tetrahedron is made of four equilateral triangles.]]
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| Equilateral triangles are found in many other geometric constructs. The intersection of circles whose centers are a radius width apart is a pair of equilateral arches, each of which can be inscribed with an equilateral triangle. They form faces of regular and uniform [[polyhedra]]. Three of the five [[Platonic solid]]s are composed of equilateral triangles. In particular, the [[Tetrahedron|regular tetrahedron]] has four equilateral triangles for faces and can be considered the three dimensional analogue of the shape. The plane can be [[Tiling by regular polygons|tiled]] using equilateral triangles giving the [[triangular tiling]].
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| ==Geometric construction==
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| [[Image:Equilateral triangle construction.svg|200px|thumb|right|Construction of equilateral triangle with compass and straightedge]]
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| An equilateral triangle is easily constructed using a [[compass and straightedge]]. Draw a straight line, and place the point of the compass on one end of the line, and swing an arc from that point to the other point of the line segment. Repeat with the other side of the line. Finally, connect the point where the two arcs intersect with each end of the line segment
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| Alternate method:
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| Draw a circle with radius ''r'', place the point of the compass on the circle and draw 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.
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| The proof that the resulting figure is an equilateral triangle is the first proposition in Book I of [[Euclid's Elements]].
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| [[File:Equilateral_Triangle_Inscribed_in_a_Circle.gif|center]]
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| ==In culture and society==
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| Equilateral triangles have frequently appeared in man made constructions:
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| *Some [[archaeological site]]s have equilateral triangles as part of their construction, for example [[Lepenski Vir]] in Serbia.
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| ==See also==
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| *[[Heronian triangle#Almost-equilateral Heronian triangles|Almost-equilateral Heronian triangle]]
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| *[[Dragon's Eye (symbol)]]
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| *[[Isosceles triangle]]
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| *[[Isoperimetric inequality]]
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| *[[Right triangle]]
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| *[[Trigonometry]]
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| ==References==
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| <references/>
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| ==External links==
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| *{{mathworld|title=Equilateral Triangle|urlname=EquilateralTriangle}}
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| {{Polygons}}
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| [[Category:Triangles]]
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| [[Category:Triangle geometry]]
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