Triangle 30654 225865739 2008-07-15T19:36:55Z THEN WHO WAS PHONE? 7456377 Reverted 1 edit by [[Special:Contributions/69.235.159.109|69.235.159.109]] identified as [[WP:VAND|vandalism]] to last revision by [[User:Riana|Riana]]. ([[WP:TW|TW]]) {{otheruses}} A '''triangle''' is one of the basic [[shape]]s of [[geometry]]: a [[polygon]] with three corners or [[wikt:vertex|vertices]] and three sides or edges which are [[line segment]]s. A triangle with vertices ''A'', ''B'', and ''C'' is denoted {{trianglenotation|ABC}}. In [[Euclidean geometry]] any three non-[[collinear]] points determine a unique triangle and a unique [[Plane (mathematics)|plane]] (i.e. two-dimensional [[Cartesian space]]). [[Image:Triangle illustration.svg|right|thumb|A triangle.]] == Types of triangles ==<!-- This section is linked from [[Pythagorean theorem]] --> Triangles can be classified according to the relative lengths of their sides: * In an '''[[equilateral triangle]]''', all sides are of equal length. An equilateral triangle is also an [[equiangular polygon]], i.e. all its internal [[angle]]s are equal&mdash;namely, 60°; it is a [[regular polygon]].<ref>{{MathWorld|title=Equilateral triangle|urlname=EquilateralTriangle}}</ref> * In an '''isosceles triangle''', two sides are of equal length (originally and conventionally limited to ''exactly'' two).<ref>Mathematicians have traditionally followed Euclid (Book 1 definition 20) in defining an isosceles triangle as having ''exactly'' two sides equal, so that equilateral triangles are excluded; but modern references tend to include equilateral triangles: [http://en.wiktionary.org/wiki/isosceles_triangle Wiktionary definition of isosceles triangle], {{MathWorld|title=Isosceles triangle|urlname=IsoscelesTriangle}}</ref> An isosceles triangle also has two equal angles: the angles opposite the two equal sides. * In a '''scalene triangle''', all sides have different lengths. The internal angles in a scalene triangle are all different.<ref>{{MathWorld|title=Scalene triangle|urlname=ScaleneTriangle}}</ref> <table align="center"><tr align="center"> <td>[[Image:Triangle.Equilateral.svg|Equilateral Triangle]]</td> <td width="125">[[Image:Triangle.Isosceles.svg|Isosceles triangle]]</td> <td>[[Image:Triangle.Scalene.svg|Scalene triangle]]</td> </tr> <tr align="center"> <td>Equilateral</td><td>Isosceles</td><td>Scalene</td> </tr> </table> Triangles can also be classified according to their internal angles, described below using [[degree (angle)|degree]]s of arc: * A '''[[Special right triangles|right triangle]]''' (or '''right-angled triangle''', formerly called a '''rectangled triangle''') has one 90° internal angle (a [[angle|right angle]]). The side opposite to the right angle is the [[hypotenuse]]; it is the longest side in the right triangle. The other two sides are the ''legs'' or '''catheti''' (singular: '''[[wiktionary:cathetus|cathetus]]''') of the triangle. * An '''oblique triangle''' has no internal angle equal to 90°. * An '''obtuse triangle''' is an oblique triangle with one internal angle larger than 90° (an [[angle|obtuse angle]]). * An '''acute triangle''' is an oblique triangle with internal angles all smaller than 90° (three [[angle|acute angle]]s). An equilateral triangle is an acute triangle, but not all acute triangles are equilateral triangles. <table align="center"> <tr align="center"> <td>[[Image:Triangle.Right.svg|Right triangle]]</td> <td width="185">[[Image:Triangle.Obtuse.svg|Obtuse triangle]]</td> <td width="185">[[Image:Triangle.Acute.svg|Acute triangle]]</td> </tr> <tr align="center"> <td>Right</td><td>Obtuse</td><td>Acute</td> </tr> <tr align="center"> <td>&nbsp;</td><td colspan="2" align="center"><math>\underbrace{\qquad \qquad \qquad \qquad \qquad \qquad}_{}</math></td> </tr> <tr align="center> <td>&nbsp;</td><td colspan="2" align="center">Oblique</td> </tr> </table> == Basic facts == Elementary facts about triangles were presented by [[Euclid]] in books 1-4 of his ''[[Euclid's Elements|Elements]]'' around 300 BCE. A triangle is a [[polygon]] and a 2-[[simplex]] (see [[polytope]]). All triangles are two-[[dimension]]al. The angles of a triangle add up to 180 degrees. An [[internal angle|exterior angle]] of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all [[convex]] polygons, the exterior angles of a triangle add up to 360 degrees. The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the [[triangle inequality]]. (In the special case of equality, two of the angles have collapsed to size zero, and the triangle has degenerated to a line segment.) Two triangles are said to be ''[[similarity (mathematics)|similar]]'' if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are [[Proportionality (mathematics)|proportional]]. This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. A few basic postulates and theorems about similar triangles: *Two triangles are similar if at least two corresponding angles are equal. *If two corresponding sides of two triangles are in proportion, and their included angles are equal, the triangles are similar. *If three sides of two triangles are in proportion, the triangles are similar. For two triangles to be congruent, each of their corresponding angles and sides must be equal (6 total). A few basic postulates and theorems about congruent triangles: *SAS Postulate: If two sides and the included angles of two triangles are correspondingly equal, the two triangles are congruent. *SSS Postulate: If every side of two triangles are correspondingly equal, the triangles are congruent. *ASA Postulate: If two angles and the included sides of two triangles are correspondingly equal, the two triangles are congruent. *AAS Theorem: If two angles and any side of two triangles are correspondingly equal, the two triangles are congruent. *Hypotenuse-Leg Theorem: If the hypotenuses and one leg of two right triangles are correspondingly equal, the triangles are congruent. Using right triangles and the concept of similarity, the [[trigonometric function]]s sine and cosine can be defined. These are functions of an [[angle]] which are investigated in [[trigonometry]]. In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known. [[Image:Pythagorean.svg|Pythagorean.svg|thumb|The Pythagorean theorem]] A central theorem is the [[Pythagorean theorem]], which states in any right triangle, the square of the length of the [[hypotenuse]] equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length ''c'', and the legs have lengths ''a'' and ''b'', then the theorem states that :<math>a^2 + b^2=c^2. \,</math> The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle. Some other facts about right triangles: * The acute angles of a right triangle are [[Complementary angles|complementary]]. * If the legs of a right triangle are equal, then the angles opposite the legs are equal, acute and complementary, and thus are both 45 degrees. By the Pythagorean theorem, the length of the hypotenuse is the length of a leg times the square root of two. * In a 30-60 right triangle, in which the acute angles measure 30 and 60 degrees, the hypotenuse is twice the length of the shorter side. * In all right triangles, the median on the hypotenuse is the half of the hypotenuse. For all triangles, angles and sides are related by the [[law of cosines]] and [[law of sines]]. == Points, lines and circles associated with a triangle == There are hundreds of different constructions that find a special point inside a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is [[Ceva's theorem]], which gives a criterion for determining when three such lines are [[concurrent lines|concurrent]]. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are [[collinear]]: here [[Menelaus' theorem]] gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. [[Image:Triangle.Circumcenter.svg|frame|right|The [[circumcenter]] is the center of a circle passing through the three vertices of the triangle.]] A [[bisection|perpendicular bisector]] of a triangle is a straight line passing through the midpoint of a side and being perpendicular to it, i.e. forming a right angle with it. The three perpendicular bisectors meet in a single point, the triangle's [[circumcenter]]; this point is the center of the [[circumcircle]], the [[circle]] passing through all three vertices. The diameter of this circle can be found from the law of sines stated above. [[Thales' theorem]] implies that if the circumcenter is located on one side of the triangle, then the opposite angle is a right one. More is true: if the circumcenter is located inside the triangle, then the triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. [[Image:Triangle.Orthocenter.svg|frame|left|The intersection of the altitudes is the [[orthocenter]].]] An [[altitude (triangle)|altitude]] of a triangle is a straight line through a vertex and perpendicular to (i.e. forming a right angle with) the opposite side. This opposite side is called the ''base'' of the altitude, and the point where the altitude intersects the base (or its extension) is called the ''foot'' of the altitude. The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the [[orthocenter]] of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. The three vertices together with the orthocenter are said to form an [[orthocentric system]]. [[Image:Triangle.Incircle.svg|frame|right|The intersection of the angle bisectors finds the center of the [[incircle]].]] An [[angle bisector]] of a triangle is a straight line through a vertex which cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the [[incenter]], the center of the triangle's [[incircle]]. The incircle is the circle which lies inside the triangle and touches all three sides. There are three other important circles, the [[excircle]]s; they lie outside the triangle and touch one side as well as the extensions of the other two. The centers of the in- and excircles form an [[orthocentric system]]. <br clear=left> [[Image:Triangle.Centroid.svg|frame|left|The intersection of the medians is the [[centroid]].]] A [[median (geometry)|median]] of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's [[centroid]]. The centroid of a stiff triangular object (cut out of a thin sheet of uniform density) is also its [[center of gravity]]: the object can be balanced it on its centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. [[Image:Triangle.NinePointCircle.svg|frame|right|[[Nine-point circle]] demonstrates a symmetry where six points lie on the edge of the triangle.]] The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle's [[nine-point circle]]. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the [[orthocenter]]. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the [[Feuerbach point]]) and the three [[excircle]]s. <br clear=left> [[Image:Triangle.EulerLine.svg|frame|left|[[Euler's line]] is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).]] The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter of the nine-point circle (red point) all lie on a single line, known as [[Euler's line]] (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. The center of the incircle is not in general located on Euler's line. If one reflects a median at the angle bisector that passes through the same vertex, one obtains a [[symmedian]]. The three symmedians intersect in a single point, the [[symmedian point]] of the triangle. <br clear=all> == Computing the area of a triangle == Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known, and simplest formula is :<math>S=\frac{1}{2}bh</math> where <math>S</math> is area, <math>b</math> is the length of the base of the triangle, and <math>h</math> is the height or altitude of the triangle. The term 'base' denotes any side, and 'height' denotes the length of a perpendicular from the point opposite the side onto the side itself. Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.<ref>{{MathWorld|title=Triangle area|urlname=TriangleArea}}</ref> ===Using vectors=== The area of a parallelogram can be calculated using [[Vector (spatial)|vectors]]. Let vectors ''AB'' and ''AC'' point respectively from A to B and from A to C. The area of parallelogram ABDC is then |''AB''&nbsp;×&nbsp;''AC''|, which is the magnitude of the [[cross product]] of vectors ''AB'' and ''AC''. |''AB''&nbsp;×&nbsp;''AC''| is equal to |''h''&nbsp;×&nbsp;''AC''|, where ''h'' represents the altitude ''h'' as a vector. The area of triangle ABC is half of this, or ''S''&nbsp;=&nbsp;½|''AB''&nbsp;×&nbsp;''AC''|. The area of triangle ABC can also be expressed in terms of [[dot product]]s as follows: :<math> \frac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\frac{1}{2} \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2} \, . </math> [[Image:Triangle.TrigArea.svg|frame|left|Applying trigonometry to find the altitude ''h''.]] ===Using trigonometry=== The height of a triangle can be found through an application of [[trigonometry]]. Using the labelling as in the image on the left, the altitude is ''h''&nbsp;=&nbsp;''a''&nbsp;sin&nbsp;γ. Substituting this in the formula ''S''&nbsp;=&nbsp;½''bh'' derived above, the area of the triangle can be expressed as: :<math>S = \frac{1}{2}ab\sin \gamma = \frac{1}{2}bc\sin \alpha = \frac{1}{2}ca\sin \beta.</math> Furthermore, since sin α = sin (''π'' - α) = sin (β + γ), and similarly for the other two angles: :<math>S = \frac{1}{2}ab\sin (\alpha+\beta) = \frac{1}{2}bc\sin (\beta+\gamma) = \frac{1}{2}ca\sin (\gamma+\alpha).</math> ===Using coordinates=== If vertex A is located at the origin (0,&nbsp;0) of a [[Cartesian coordinate system]] and the coordinates of the other two vertices are given by B&nbsp;=&nbsp;(''x''<sub>B</sub>,&nbsp;''y''<sub>B</sub>) and C&nbsp;=&nbsp;(''x''<sub>C</sub>,&nbsp;''y''<sub>C</sub>), then the area ''S'' can be computed as ½ times the [[absolute value]] of the [[determinant]] :<math>S=\frac{1}{2}\left|\det\begin{pmatrix}x_B & x_C \\ y_B & y_C \end{pmatrix}\right| = \frac{1}{2}|x_B y_C - x_C y_B|. </math> For three general vertices, the equation is: :<math>S=\frac{1}{2} \left| \det\begin{pmatrix}x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1\end{pmatrix} \right| = \frac{1}{2} \big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A \big|. </math> In three dimensions, the area of a general triangle {A&nbsp;=&nbsp;(''x''<sub>A</sub>,&nbsp;''y''<sub>A</sub>,&nbsp;''z''<sub>A</sub>), B&nbsp;=&nbsp;(''x''<sub>B</sub>,&nbsp;''y''<sub>B</sub>,&nbsp;''z''<sub>B</sub>) and C&nbsp;=&nbsp;(''x''<sub>C</sub>,&nbsp;''y''<sub>C</sub>,&nbsp;''z''<sub>C</sub>)} is the [[Pythagorean sum]] of the areas of the respective projections on the three principal planes (i.e. ''x'' = 0, ''y'' = 0 and ''z'' = 0): :<math>S=\frac{1}{2} \sqrt{ \left( \det\begin{pmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 + \left( \det\begin{pmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 }. </math> ===Using Heron's formula=== The shape of the triangle is determined by the lengths of the sides alone. Therefore the area ''S'' also can be derived from the lengths of the sides. By [[Heron's formula]]: :<math>S = \sqrt{s(s-a)(s-b)(s-c)}</math> where ''s''&nbsp;=&nbsp;½&nbsp;(''a''&nbsp;+&nbsp;''b''&nbsp;+&nbsp;''c'') is the '''semiperimeter''', or half of the triangle's perimeter. An equivalent way of writing Heron's formula is :<math> S = \frac{1}{4} \sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}.</math> == Non-planar triangles == A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in noneuclidean geometries are [[spherical triangle]]s in [[spherical geometry]] and [[hyperbolic triangle]]s in [[hyperbolic geometry]]. While all regular, planar (two dimensional) triangles contain angles that add up to 180°, there are cases in which the angles of a triangle can be greater than or less than 180°. In curved figures, a triangle on a negatively curved figure ("saddle") will have its angles add up to less than 180° while a triangle on a positively curved figure ("sphere") will have its angles add up to more than 180°. Thus, if one were to draw a giant triangle on the surface of the Earth, one would find that the sum of its angles were greater than 180°. ==See also== <div style="-moz-column-count:3; column-count:3;"> *[[List of triangle topics]] *[[Triangular number]] *[[Special right triangles]] *[[Fermat point]] *[[Hadwiger-Finsler inequality]] *[[Pedoe's inequality]] *[[Ono's inequality]] *[[Lester's theorem]] *[[Congruence (geometry)]] *[[Pythagorean theorem]] *[[Law of sines]] *[[Law of cosines]] *[[Law of tangents]] *[[Triangulation (topology)]] of a [[manifold]] *[[Triangulated category]] *[[Inertia tensor of triangle]] *[[A-frame for hang gliders, trikes, and ultralights]] </div> ==References== <references/> == External links == {{CommonsCat|Triangles}} * [http://www.geometryatlas.com/categories/Triangles Triangle Formulas] at Geometry Atlas. *[http://ostermiller.org/calc/triangle.html Triangle Calculator] - solves for remaining sides and angles when given three sides or angles, supports degrees and radians. * Clark Kimberling: [http://faculty.evansville.edu/ck6/encyclopedia/ETC.html Encyclopedia of triangle centers]. Lists some 3200 interesting points associated with any triangle. * Christian Obrecht: [http://www.eukleides.org/ Eukleides]. Software package for creating illustrations of facts about triangles and other theorems in Euclidean geometry. * [http://www.apronus.com/geometry/triangle.htm Proof that the sum of the angles in a triangle is 180 degrees] * [http://www.btinternet.com/~se16/hgb/triangle.htm Area of a triangle - 7 different ways] * [http://www.mathopenref.com/tocs/triangletoc.html Triangle definition pages] with interactive applets that are also useful in a classroom setting. * [http://www.mathopenref.com/tocs/constructionstoc.html Animated demonstrations] of triangle constructions using compass and straightedge. * [http://agutie.homestead.com/files/triangles_1.htm Triangles: Theorems and Problems.] Interactive illustrations at [[Geometry from the Land of the Incas]]. * [http://mathworld.wolfram.com/Triangle.html Many things about triangles] * [http://www.allmathwords.org/triangle.html All Math Words Encyclopedia for grades 7-10.] [[Category:Polygons]] [[Category:Triangles]] [[Category:Triangle geometry]] {{Link FA|km}} {{Link FA|pt}} [[ar:مثلث]] [[an:Trianglo]] [[ast:Triángulu]] [[ay:Mujina]] [[az:Üçbucaqlar]] [[bn:ত্রিভুজ]] [[zh-min-nan:Saⁿ-kak-hêng]] [[be:Трохвугольнік]] [[be-x-old:Трыкутнік]] [[bs:Trougao]] [[bg:Триъгълник]] [[ca:Triangle]] [[cv:Виç кĕтеслĕх]] [[cs:Trojúhelník]] [[co:Triangulu]] [[cy:Triongl]] [[da:Trekant]] [[de:Dreieck]] [[et:Kolmnurk]] [[el:Τρίγωνο]] [[es:Triángulo]] [[eo:Triangulo]] [[eu:Hiruki]] [[fa:مثلث]] [[fr:Triangle]] [[gl:Triángulo]] [[zh-classical:三角形]] [[ko:삼각형]] [[hr:Trokut]] [[io:Triangulo]] [[id:Segitiga]] [[is:Þríhyrningur]] [[it:Triangolo]] [[he:משולש]] [[ka:სამკუთხედი]] [[sw:Pembetatu]] [[ht:Triyang]] [[ku:Sêgoşe]] [[la:Triangulum]] [[lv:Trīsstūris]] [[lt:Trikampis]] [[li:Driehook]] [[hu:Háromszög]] [[mk:Триаголник]] [[ml:ത്രികോണം]] [[mr:त्रिकोण]] [[ms:Segi tiga]] [[mn:Гурвалжин]] [[nl:Driehoek (meetkunde)]] [[new:त्रिकोण]] [[ja:三角形]] [[no:Trekant]] [[nn:Trekant]] [[nrm:Trian]] [[km:ត្រីកោណ]] [[pl:Trójkąt]] [[pt:Triângulo]] [[ro:Triunghi]] [[qu:Kimsak'uchu]] [[ru:Треугольник]] [[sco:Triangle]] [[simple:Triangle]] [[sk:Trojuholník]] [[sl:Trikotnik]] [[sr:Троугао]] [[sh:Trokut]] [[fi:Kolmio]] [[sv:Triangel]] [[ta:முக்கோணம்]] [[te:త్రిభుజం]] [[th:รูปสามเหลี่ยม]] [[vi:Tam giác]] [[tg:Секунҷа]] [[tr:Üçgen]] [[uk:Трикутник]] [[ur:مثلث]] [[vls:Drieoek]] [[yi:דרייעק]] [[yo:Anígunmẹ́ta]] [[zh-yue:三角形]] [[zh:三角形]]