Triangle

This article is about the basic geometric shape. For other uses, seeTriangle (disambiguation).

Atriangleis apolygonwith three corners and three sides, one of the basicshapesingeometry.The corners, also calledvertices,are zero-dimensionalpointswhile the sides connecting them, also callededges,are one-dimensionalline segments.A triangle has threeinternal angles,each one bounded by a pair of adjacent edges; thesum of angles of a trianglealways equals astraight angle(180 degrees or π radians). The triangle is aplane figureand its interior is aplanar region.Sometimes an arbitrary edge is chosen to be thebase,in which case the opposite vertex is called theapex;the shortest segment between the base and apex is theheight.Thearea of a triangleequals one-half the product of height and base length.

Triangle
Edgesandvertices	3
Schläfli symbol	{3} (for equilateral)
Area	various methods; see below

InEuclidean geometry,any two points determine a unique line segment situated within a uniquestraight line,and any three points that do notall lie on the same straight linedetermine a unique triangle situated within a unique flatplane.More generally, four points inthree-dimensional Euclidean spacedetermine atetrahedron.

Innon-Euclidean geometries,three "straight" segments (having zerocurvature) also determine a triangle, for instance, aspherical triangleorhyperbolic triangle.Ageodesic triangleis a region of a general two-dimensionalsurfaceenclosed by three sides that are straight relative to the surface (geodesics). Acurvilineartriangle is a shape with threecurvedsides, for instance, acircular trianglewithcircular-arcsides. This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.

Triangles are classified into different types based on their angles and the lengths of their sides. Relations between angles and side lengths are a major focus oftrigonometry.In particular, thesine, cosine, and tangent functionsrelate side lengths and angles inright triangles.

Definition, terminology, and types

A triangle is a figure consisting of three line segments, each of whose endpoints are connected.This forms a polygon with three sides and three angles. The terminology for categorizing triangles is more than two thousand years old, having been defined in Book One of Euclid'sElements.The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations.

Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is anequilateral triangle,a triangle with two sides having the same length is anisosceles triangle,and a triangle with three different-length sides is ascalene triangle.A triangle in which one of the angles is aright angleis aright triangle,a triangle in which all of its angles are less than that angle is anacute triangle,and a triangle in which one of it angles is greater than that angle is anobtuse triangle.These definitions date back at least toEuclid.

• 
  Equilateral triangle
• 
  Isosceles triangle
• 
  Scalene triangle

• 
  Right triangle
• 
  Acute triangle
• 
  Obtuse triangle

Appearances

All types of triangles are commonly found in real life. In man-made construction, the isosceles triangles may be found in the shape ofgablesandpediments,and the equilateral triangle can be found in the yield sign.The faces of theGreat Pyramid of Gizaare sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead.Other appearances are inheraldicsymbols as in theflag of Saint Luciaandflag of the Philippines.

Atriangular bipyramidcan be constructed by attaching twotetrahedra.This polyhedron can be said to be asimplicial polyhedronbecause all of its faces are triangles. More specifically, when the faces are equilateral, it is categorized as adeltahedron.

Triangles also appear in three-dimensional objects. Apolyhedronis a solid whose boundary is covered by flatpolygonalsknown as the faces, sharp corners known as the vertices, and line segments known as the edges. Polyhedra in some cases can be classified, judging from the shape of their faces. For example, when polyhedra have all equilateral triangles as their faces, they are known asdeltahedra.Antiprismshave alternating triangles on their sides.Pyramidsandbipyramidsare polyhedra with polygonal bases and triangles for lateral faces; the triangles are isosceles whenever they are right pyramids and bipyramids. TheKleetopeof a polyhedron is a new polyhedron made by replacing each face of the original with a pyramid, and so the faces of a Kleetope will be triangles.More generally, triangles can be found in higher dimensions, as in the generalized notion of triangles known as thesimplex,and thepolytopeswith triangularfacetsknown as thesimplicial polytopes.

Properties

Points, lines, and circles associated with a triangle

Main article:Encyclopedia of Triangle Centers

Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically 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 points isCeva's theorem,which gives a criterion for determining when three such lines areconcurrent.Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points arecollinear;hereMenelaus' theoremgives a useful general criterion.In this section, just a few of the most commonly encountered constructions are explained.

Thecircumcenteris the center of a circle passing through the three vertices of the triangle; the intersection of the altitudes is theorthocenter.The intersection of the angle bisectors is the center of theincircle.

Aperpendicular bisectorof a side of a triangle is a straight line passing through themidpointof the side and being perpendicular to it, forming a right angle with it.The three perpendicular bisectors meet in a single point, the triangle'scircumcenter;this point is the center of thecircumcircle,the circle passing through all three vertices.Thales' theoremimplies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle.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.

Analtitudeof a triangle is a straight line through a vertex and perpendicular to 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 theorthocenterof the triangle.The orthocenter lies inside the triangle if and only if the triangle is acute.

Nine-point circledemonstrates a symmetry where six points lie on the edge of the triangle.Euler's lineis a straight line through the orthocenter (blue), the center of the nine-point circle (red), centroid (orange), and circumcenter (green).

Anangle bisectorof a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, theincenter,which is the center of the triangle'sincircle.The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, theexcircles;they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form anorthocentric system.The midpoints of the three sides and the feet of the three altitudes all lie on a single circle, the triangle'snine-point circle.The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and theorthocenter.The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at theFeuerbach point) and the threeexcircles.The orthocenter (blue point), the center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known asEuler'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.Generally, the incircle's center is not located on Euler's line.

The incircle of a triangle, and the intersection of the medians known as thecentroid

Amedianof a triangle is a straight line through avertexand themidpointof the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle'scentroidor geometric barycenter. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also itscenter of mass:the object can be balanced on its centroid in a uniform gravitational field.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. If one reflects a median in the angle bisector that passes through the same vertex, one obtains asymmedian.The three symmedians intersect in a single point, thesymmedian pointof the triangle.

Angles

The measures of the interior angles of the triangle always add up to 180 degrees (same color to point out they are equal).

Thesum of the measures of the interior angles of a triangleinEuclidean spaceis always 180 degrees.This fact is equivalent to Euclid'sparallel postulate.This allows the determination of the measure of the third angle of any triangle, given the measure of two angles.Anexterior angleof a triangle is an angle that is a linear pair (and hencesupplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is theexterior angle theorem.The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has.

Another relation between the internal angles and triangles creates a new concept oftrigonometric functions.The primary trigonometric functions aresine and cosine,as well as the other functions. They can be defined as theratio between any two sides of a right triangle.In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use thelaw of sinesand thelaw of cosines.

Any three angles that add to 180° can be the internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (Adegenerate triangle,whose vertices arecollinear,has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention.) The conditions for three angles$\ Alpha$,$\beta$,and$\gamma$,each of them between 0° and 180°, to be the angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles$\ Alpha$,$\beta$,and$\gamma$existsif and only if $$\cos ^{2}\ Alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\ Alpha )\cos(\beta )\cos(\gamma )=1.$$

Similarity and congruence

This diagram illustrates the geometric principle of angle-angle-side triangle congruence: given triangle ABC and triangle A'B'C', triangle ABC is congruent with triangle A'B'C' if and only if: angle CAB is congruent with angle C'A'B', and angle ABC is congruent with angle A'B'C', and BC is congruent with B'C'. Notehatch marksare used here to show angle and side equalities.

Two triangles are said to besimilar,if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity.

Some basictheoremsabout similar triangles are:

• If and only ifone pair of internal angles of two triangles have the same measure as each other, and another pair also have the same measure as each other, the triangles are similar.
• If and only if one pair of corresponding sides of two triangles are in the same proportion as another pair of corresponding sides, and their included angles have the same measure, then the triangles are similar.(Theincluded anglefor any two sides of a polygon is the internal angle between those two sides.)
• If and only if three pairs of corresponding sides of two triangles are all in the same proportion, then the triangles are similar.

Two triangles that arecongruenthave exactly the same size and shape. All pairs of congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. This is a total of six equalities, but three are often sufficient to prove congruence.

Some individuallynecessary and sufficient conditionsfor a pair of triangles to be congruent are:

• SAS Postulate: Two sides in a triangle have the same length as two sides in the other triangle, and the included angles have the same measure.
• ASA: Two interior angles and the side between them in a triangle have the same measure and length, respectively, as those in the other triangle. (This is the basis ofsurveying by triangulation.)
• SSS: Each side of a triangle has the same length as the corresponding side of the other triangle.
• AAS: Two angles and a corresponding (non-included) side in a triangle have the same measure and length, respectively, as those in the other triangle. (This is sometimes referred to asAAcorrSand then includes ASA above.)

Area

Main article:Area of a triangle

The area formula for a triangle can be proven by cutting two copies of the triangle into pieces and rearranging them into a rectangle.

In the Euclidean plane,areais defined by comparison with a square of side length⁠$1$⁠,which has area 1. There are several ways to calculate the area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side⁠$b$⁠(the base) times the corresponding altitude⁠$h$⁠: $$T={\tfrac {1}{2}}bh.$$

This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base⁠$b$⁠and height⁠$h$⁠.

Applying trigonometry to find the altitudeh

If two sides⁠$a$⁠and⁠$b$⁠and their included angle$\gamma$are known, then the altitude can be calculated using trigonometry,⁠$h=a\sin(\gamma )$⁠,so the area of the triangle is: $$T={\tfrac {1}{2}}ab\sin \gamma.$$

Heron's formula,named afterHeron of Alexandria,is a formula for finding the area of a triangle from the lengths of its sides$a$,$b$,$c$.Letting$s={\tfrac {1}{2}}(a+b+c)$be thesemiperimeter, $$T={\sqrt {s(s-a)(s-b)(s-c)}}.$$

Orange triangles△ABCshare a baseABand area. The locus of their apexCis a line (dashed green) parallel to the base. This is the Euclidean version ofLexell's theorem.

Because the ratios between areas of shapes in the same plane are preserved byaffine transformations,the relative areas of triangles in anyaffine planecan be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with the same base andoriented areahas its apex (the third vertex) on a line parallel to the base, and their common area is half of that of aparallelogramwith the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid'sElements.

Givenaffine coordinates(such asCartesian coordinates)⁠$(x_{A},y_{A})$⁠,⁠$(x_{B},y_{B})$⁠,⁠$(x_{C},y_{C})$⁠for the vertices of a triangle, its relative oriented area can be calculated using theshoelace formula,

$${\begin{aligned}T&={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}\\y_{A}&y_{B}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{C}&x_{A}\\y_{C}&y_{A}\end{vmatrix}}\\&={\tfrac {1}{2}}(x_{A}y_{B}-x_{B}y_{A}+x_{B}y_{C}-x_{C}y_{B}+x_{C}y_{A}-x_{A}y_{C}),\end{aligned}}$$

where$|\cdot |$is thematrix determinant.

Possible side lengths

Main article:Triangle inequality

Thetriangle inequalitystates that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side.Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality.The sum of two side lengths can equal the length of the third side only in the case of adegenerate triangle,one with collinear vertices.

Rigidity

Main article:Structural rigidity

Rigidity of a triangle and square

Unlike a rectangle, which may collapse into aparallelogramfrom pressure to one of its points,triangles are sturdy because specifying the lengths of all three sides determines the angles.Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports the other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense.

Triangles are strong in terms of rigidity, but while packed in atessellatingarrangement triangles are not as strong ashexagonsunder compression (hence the prevalence of hexagonal forms innature). Tessellated triangles still maintain superior strength forcantilevering,however, which is why engineering makes use oftetrahedral trusses.

Triangulation

Triangulation in a simple polygon

Triangulationmeans the partition of any planar object into a collection of triangles. For example, inpolygon triangulation,a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon.In the case of asimple polygonwith$n$sides, there are$n-2$triangles that are separated by$n-3$diagonals. Triangulation of a simple polygon has a relationship to theear,a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. Thetwo ears theoremstates that every simple polygon that is not itself a triangle has at least two ears.

Location of a point

One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in theCartesian plane,and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.

Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to a similar triangle:

• Trilinear coordinatesspecify the relative distances of a point from the sides, so that coordinates$x:y:z$indicate that the ratio of the distance of the point from the first side to its distance from the second side is$x:y$,etc.
• Barycentric coordinatesof the form$\ Alpha:\beta:\gamma$specify the point's location by the relative weights that would have to be put on the three vertices in order to balance the otherwise weightless triangle on the given point.

Related figures

Figures inscribed in a triangle

As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a uniqueSteiner inellipsewhich is interior to the triangle and tangent at the midpoints of the sides.Marden's theoremshows how to find thefoci of this ellipse.This ellipse has the greatest area of any ellipse tangent to all three sides of the triangle. TheMandart inellipseof a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle$ABC$,let the foci be$P$and$Q$,then: $${\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.$$

Thepedal triangleandGergonne triangle

From an interior point in a reference triangle, the nearest points on the three sides serve as the vertices of thepedal triangleof that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called themidpoint triangleor medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to the reference triangle.

Theintouch triangleof a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle.Theextouch triangleof a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended).

Every acute triangle has three inscribed squares (squares in its interior such that all four of a square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only twodistinctinscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length$q_{a}$and the triangle has a side of length$a$,part of which side coincides with a side of the square, then$q_{a}$,$a$,$h_{a}$from the side$a$,and the triangle's area$T$are related according to$$q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.$$The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when$a^{2}=2T$,$q=a/2$,and the altitude of the triangle from the base of length$a$is equal to$a$.The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is$2{\sqrt {2}}/3$.Both of these extreme cases occur for the isosceles right triangle.

The Lemoine hexagon inscribed in a triangle

TheLemoine hexagonis acyclic hexagonwith vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through itssymmedian point.In either itssimple form or its self-intersecting form,the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle.

Everyconvex polygonwith area$T$can be inscribed in a triangle of area at most equal to$2T$.Equality holds only if the polygon is aparallelogram.

Figures circumscribed about a triangle

The circumscribed circle tangent to a triangle and theSteiner circumellipse

Thetangential triangleof a reference triangle (other than a right triangle) is the triangle whose sides are on thetangent linesto the reference triangle's circumcircle at its vertices.

As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a uniqueSteiner circumellipse,which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area.

TheKiepert hyperbolais uniqueconicthat passes through the triangle's three vertices, its centroid, and its circumcenter.

Of all triangles contained in a givenconvex polygon,one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon.

Miscellaneous triangles

Circular triangles

Main article:Circular triangle

Circular triangles with a mixture of convex and concave edges

Acircular triangleis a triangle with circulararcedges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward).The intersection of threedisksforms a circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is aReuleaux triangle,which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by theMohr–Mascheroni theorem.Alternatively, it can be constructed by rounding the sides of an equilateral triangle.

A special case of concave circular triangle can be seen in apseudotriangle.A pseudotriangle is asimply-connectedsubset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called thecusp points.Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks andbitangent lines,a process known as pseudo-triangulation. For$n$disks in a pseudotriangle, the partition gives$2n-2$pseudotriangles and$3n-3$bitangent lines.Theconvex hullof any pseudotriangle is a triangle.

Triangle in non-planar space

Main articles:Hyperbolic triangleandSpherical triangle

Hyperbolic triangleandspherical triangle

A non-planar triangle is a triangle not included inEuclidean space,roughly speaking a flat space. This means triangles may also be discovered in several spaces, as inhyperbolic spaceandspherical geometry.A triangle in hyperbolic space is called ahyperbolic triangle,and it can be obtained by drawing on a negatively curved surface, such as asaddle surface.Likewise, a triangle in spherical geometry is called aspherical triangle,and it can be obtained by drawing on a positively curved surface such as asphere.

The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above, the internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°.In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. ByGirard's theorem,the sum of the angles of a triangle on a sphere is$180^{\circ }\times (1+4f)$,where$f$is the fraction of the sphere's area enclosed by the triangle.

In more general spaces, there arecomparison theoremsrelating the properties of a triangle in the space to properties of a corresponding triangle in a model space like hyperbolic or elliptic space.For example, aCAT(k) spaceis characterized by such comparisons.

Fractal geometry

Fractalshapes based on triangles include theSierpiński gasketand theKoch snowflake.

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External links

• Ivanov, A.B. (2001) [1994]."Triangle".Encyclopedia of Mathematics.EMS Press.
• Clark Kimberling:Encyclopedia of triangle centers.Lists some 5200 interesting points associated with any triangle.