This list of

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

topics includes things related to the geometric shape, either abstractly, as in idealizations studied by geometers, or in triangular arrays such as

Pascal's triangle In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Bla ...

or triangular matrices, or concretely in physical space. It does not include metaphors like

love triangle A love triangle is a scenario or circumstance, usually depicted as a rivalry, in which two people are pursuing or involved in a romantic relationship with one person, or in which one person in a romantic relationship with someone is simultaneo ...

in which the word has no reference to the geometric shape.

Geometry

Triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

* Acute and obtuse triangles *

Altitude (triangle) In geometry, an altitude of a triangle is a line segment through a given Vertex (geometry), vertex (called ''apex (geometry), apex'') and perpendicular to a line (geometry), line containing the side or edge (geometry), edge opposite the apex. Th ...

* Area bisector of a triangle * Angle bisector of a triangle * Angle bisector theorem * Apollonius point * Apollonius' theorem * Automedian triangle * Barrow's inequality * Barycentric coordinates (mathematics) * Bernoulli's quadrisection problem * Brocard circle * Brocard points * Brocard triangle * Carnot's theorem (conics) * Carnot's theorem (inradius, circumradius) * Carnot's theorem (perpendiculars) * Catalogue of Triangle Cubics *

Centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...

* Ceva's theorem * Cevian * Circumconic and inconic * Circumscribed circle * Clawson point * Cleaver (geometry) *

Congruence (geometry) In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be ...

* Congruent isoscelizers point * Contact triangle * Conway triangle notation * CPCTC *

Delaunay triangulation In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles whose circumcircles do not contain any of the points; that is, each circumcircle has its gen ...

* de Longchamps point * Desargues' theorem * Droz-Farny line theorem * Encyclopedia of Triangle Centers * Equal incircles theorem * Equal parallelians point * Equidissection *

Equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

* Euler's line * Euler's theorem in geometry * Erdős–Mordell inequality * Exeter point * Exterior angle theorem * Fagnano's problem * Fermat point * Fermat's right triangle theorem * Fuhrmann circle * Fuhrmann triangle * Geometric mean theorem * GEOS circle * Gergonne point * Golden triangle (mathematics) * Gossard perspector * Hadwiger–Finsler inequality * Heilbronn triangle problem * Heptagonal triangle * Heronian triangle * Heron's formula * Hofstadter points *

Hyperbolic triangle In hyperbolic geometry, a hyperbolic triangle is a triangle in the hyperbolic plane. It consists of three line segments called ''sides'' or ''edges'' and three point (geometry), points called ''angles'' or ''vertices''. Just as in the Euclidea ...

(non-Euclidean geometry) * Hypotenuse * Incircle and excircles of a triangle * Inellipse * Integer triangle * Isodynamic point * Isogonal conjugate * Isoperimetric point *

Isosceles triangle In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...

* Isosceles triangle theorem * Isotomic conjugate * Isotomic lines * Jacobi point * Japanese theorem for concyclic polygons * Johnson circles * Kepler triangle * Kobon triangle problem * Kosnita's theorem * Leg (geometry) * Lemoine's problem * Lester's theorem * List of triangle inequalities * Mandart inellipse *

Maxwell's theorem (geometry) Maxwell's theorem is the following statement about triangles in the plane. The theorem is named after the physicist James Clerk Maxwell (1831–1879), who proved it in his work on reciprocal figures, which are of importance in statics. Refere ...

* Medial triangle * Median (geometry) *

Menelaus' theorem In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle , and a Transversal (geometry), transversal line that crosses at points respectively, wi ...

* Miquel's theorem * Mittenpunkt * Modern triangle geometry * Monsky's theorem * Morley centers *

Morley triangle In plane geometry, 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, called the first Morley triangle or simply the Morley tria ...

* Morley's trisector theorem * Musselman's theorem * Nagel point * Napoleon points * Napoleon's theorem * Nine-point circle * Nine-point hyperbola *

One-seventh area triangle In plane geometry, a triangle ''ABC'' contains a triangle having one-seventh of the area of ''ABC'', which is formed as follows: the sides of this triangle lie on cevians ''p, q, r'' where :''p'' connects ''A'' to a point on ''BC'' that is one-th ...

Orthocenter The orthocenter of a triangle, usually denoted by , is the point (geometry), point where the three (possibly extended) altitude (triangle), altitudes intersect. The orthocenter lies inside the triangle if and only if the triangle is acute trian ...

Orthocentric system In geometry, an orthocentric system is a set (mathematics), set of four point (geometry), points on a plane (mathematics), plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through ...

* Orthocentroidal circle * Orthopole * Pappus' area theorem * Parry point * Pedal triangle * Perimeter bisector of a triangle * Perpendicular bisectors of triangle sides *

Polar circle (geometry) In geometry, the polar circle of a triangle is the circle whose center is the triangle's orthocenter and whose squared radius is where denote both the triangle's vertex (geometry), vertices and the angle measures at those vertices; is the or ...

* Pompeiu's theorem * Pons asinorum *

Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...

** Inverse Pythagorean theorem * Reuleaux triangle * Regiomontanus * Regiomontanus' angle maximization problem * Reuschle's theorem *

Right triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle ( turn or 90 degrees). The side opposite to the right angle i ...

Routh's theorem In geometry, Routh's theorem determines the ratio of areas between a given triangle and a triangle formed by the pairwise intersections of three cevians. The theorem states that if in triangle ABC points D, E, and F lie on segments BC, CA, and ...

Scalene triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimensional ...

* Schwarz triangle * Schiffler's theorem * Sierpinski triangle (fractal geometry) *

Similarity (geometry) In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling (geometry), scaling (enlarging or ...

Spiral of Theodorus In geometry, the spiral of Theodorus (also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene. Construction The spiral ...

* Splitter (geometry) * Steiner circumellipse * Steiner inellipse * Steiner–Lehmus theorem * Stewart's theorem * Steiner point * Symmedian *

Tangential triangle In geometry, the tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertex (geometry), vertices. Thus ...

* Tarry point * Ternary plot *

Thales' theorem In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as pa ...

* Thomson cubic * Triangle center * Triangle conic *

Triangle group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triang ...

Triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of Degeneracy (mathematics)#T ...

* Triangular bipyramid *

Triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

* Triangular pyramid * Triangular tiling *

Triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle m ...

* Trilinear coordinates * Trilinear polarity * Trisected perimeter point *

Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from ''any'' interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. It is a theorem commonly employed in ...

* Yff center of congruence

Trigonometry

* Differentiation of trigonometric functions * Exact trigonometric constants *

History of trigonometry Early study of triangles can be traced to Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics during the 2nd millennium BC. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric funct ...

Inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a functi ...

Law of cosines In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides , , and , opposite respective angles , , and (see ...

* Law of cotangents *

Law of sines In trigonometry, the law of sines (sometimes called the sine formula or sine rule) is a mathematical equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\ ...

* Law of tangents * List of integrals of inverse trigonometric functions * List of integrals of trigonometric functions *

List of trigonometric identities In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involvin ...

* Mollweide's formula * Outline of trigonometry * Rational trigonometry *

Sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite th ...

* Solution of triangles *

Spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the edge (geometry), sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, ge ...

Trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all ...

* Trigonometric substitution *

Trigonometric tables In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering. The calculation of mathematical tables ...

Trigonometry Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...

* Uses of trigonometry

Applied mathematics

* De Finetti diagram * Triangle mesh * Nonobtuse mesh

Resources

* '' Encyclopedia of Triangle Centers'' * '' Pythagorean Triangles'' * '' The Secrets of Triangles''

Algebra

Triangular matrix In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are z ...

* (2,3,7) triangle group

Number theory

Triangular arrays of numbers

* Bell numbers * Boustrophedon transform * Eulerian number * Floyd's triangle * Lozanić's triangle *

Narayana number In combinatorics, the Narayana numbers \operatorname(n, k), n \in \mathbb^+, 1 \le k \le n form a triangular array of natural numbers, called the Narayana triangle, that occur in various Combinatorial enumeration, counting problems. They are named ...

* Rencontres numbers *

Romberg's method In numerical analysis, Romberg's method is used to estimate the Integral, definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate ...

* Stirling numbers of the first kind * Stirling numbers of the second kind *

Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in ...

* Triangular pyramidal number The (incomplete) Bell polynomials from a triangular array of polynomials (see also

Polynomial sequence In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in ...

Integers in triangle geometry

* Heronian triangle * Integer triangle *

Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are a Py ...

* Eisenstein triple

Geography

Bermuda Triangle The Bermuda Triangle, also known as the Devil's Triangle, is a loosely defined region in the North Atlantic Ocean, roughly bounded by Florida, Bermuda, and Puerto Rico. Since the mid-20th century, it has been the focus of an urban legend sug ...

* Historic Triangle * Lithium Triangle * Parliamentary Triangle, Canberra *

Research Triangle The Research Triangle, or simply The Triangle, are both common nicknames for a metropolitan area in the Piedmont (United States), Piedmont region of the U.S. state of North Carolina. Anchored by the cities of Raleigh, North Carolina, Raleigh an ...

* Sunni Triangle *

Triangular trade Triangular trade or triangle trade is trade between three ports or regions. Triangular trade usually evolves when a region has export commodities that are not required in the region from which its major imports come. It has been used to offset ...

* Rhubarb Triangle

Anatomy

* Submandibular triangle * Triangle choke * Arm triangle choke *

Submental triangle The submental triangle (or suprahyoid triangle) is a division of the anterior triangle of the neck. Boundaries It is limited to: * Lateral (away from the midline), formed by the anterior belly of the digastricus * Medial (towards the midline), ...

* Carotid triangle * Clavipectoral triangle * Inguinal triangle * Codman triangle

Artifacts

Black triangle Black triangle may refer to: Places * Black Triangle (region), across Germany, Poland and the Czech Republic, long characterized by extremely high levels of pollution * Black triangle, the nickname given to the area south of Montreal affected by a ...

Triangle (musical instrument) The triangle, or musical triangle, is a musical instrument in the percussion family, classified as an idiophone in the Hornbostel-Sachs classification system. Triangles are made from a variety of metals including aluminum, beryllium copper, ...

Triangular prism (optics) In optics, a dispersive prism is an optical prism that is used to disperse light, that is, to separate light into its spectral components (the colors of the rainbow). Different wavelengths (colors) of light will be deflected by the prism at ...

* Triquetra

Symbols

* Eye of Providence * Valknut * Shield of the Trinity {{DEFAULTSORT:Triangle Topics Topics Topics Outlines of mathematics and logic Outlines Lists of topics Lists of shapes