This list of

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although ot ...

triangular matrices 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 ...

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

love triangle A love triangle or eternal 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 so ...

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

Geometry

Triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

Acute and obtuse triangles An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's ang ...

* Altern base *

Altitude (triangle) In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

* Area bisector of a triangle * Angle bisector of a triangle *

Angle bisector theorem In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the ...

Apollonius point In Euclidean geometry, the Apollonius point is a triangle center designated as ''X''(181) in Clark Kimberling's Encyclopedia of Triangle Centers (ETC). It is defined as the point of concurrence of the three line segments joining each vertex of the ...

Apollonius' theorem In geometry, Apollonius's theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that "the sum of the squares of any two sides of any triangle equals twice the square on half the third side, to ...

Automedian triangle In plane geometry, an automedian triangle is a triangle in which the lengths of the three medians (the line segments connecting each vertex to the midpoint of the opposite side) are proportional to the lengths of the three sides, in a different or ...

Barrow's inequality In geometry, Barrow's inequality is an inequality relating the distances between an arbitrary point within a triangle, the vertices of the triangle, and certain points on the sides of the triangle. It is named after David Francis Barrow. Stateme ...

Barycentric coordinates (mathematics) In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The baryc ...

* Bernoulli's quadrisection problem *

Brocard circle In geometry, the Brocard circle (or seven-point circle) is a circle derived from a given triangle. It passes through the circumcenter and symmedian of the triangle, and is centered at the midpoint of the line segment joining them (so that this s ...

Brocard points In geometry, Brocard points are special points within a triangle. They are named after Henri Brocard (1845–1922), a French mathematician. Definition In a triangle ''ABC'' with sides ''a'', ''b'', and ''c'', where the vertices are labeled '' ...

Brocard triangle In geometry, the Brocard triangle of a triangle is a triangle formed by the intersection of lines from a vertex to its corresponding Brocard point and a line from another vertex to its corresponding Brocard point and the other two points construc ...

Carnot's theorem (conics) Carnot's theorem (named after Lazare Carnot) describes a relation between conic sections and triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'' ...

Carnot's theorem (inradius, circumradius) In Euclidean geometry, Carnot's theorem states that the sum of the signed distances from the circumcenter ''D'' to the sides of an arbitrary triangle ''ABC'' is :DF + DG + DH = R + r,\ where ''r'' is the inradius and ''R'' is the circumradiu ...

Carnot's theorem (perpendiculars) Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection. The theorem can also be thought of as ...

Catalogue of Triangle Cubics The Catalogue of Triangle Cubics is an online resource containing detailed information about more than 1200 cubic curves in the plane of a reference triangle. The resource is maintained by Bernard Gilbert. Each cubic in the resource is assigned a ...

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 surface of the figure. The same definition extends to any ob ...

Ceva's theorem In Euclidean geometry, Ceva's theorem is a theorem about triangles. Given a triangle , let the lines be drawn from the vertices to a common point (not on one of the sides of ), to meet opposite sides at respectively. (The segments are kn ...

Cevian In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovann ...

Circumconic and inconic In Euclidean geometry, a circumconic is a conic section that passes through the three vertices of a triangle, and an inconic is a conic section inscribed in the sides, possibly extended, of a triangle.Weisstein, Eric W. "Inconic." From MathWorld ...

Circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

Clawson point The Clawson point is a special point in a planar triangle defined by the trilinear coordinates \tan(\alpha):\tan(\beta):\tan(\gamma) ( Kimberling number X(19)), where \alpha, \beta, \gamma are the interior angles at the triangle vertices A, B, C. I ...

Cleaver (geometry) In geometry, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. They are not to be confused with splitters, which also bisect the perimeter, but w ...

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 In geometry, the Conway triangle notation, named after John Horton Conway, allows trigonometric functions of a triangle to be managed algebraically. Given a reference triangle whose sides are ''a'', ''b'' and ''c'' and whose corresponding internal a ...

* CPCTC *

Delaunay triangulation In mathematics and computational geometry, a Delaunay triangulation (also known as a Delone triangulation) for a given set P of discrete points in a general position is a triangulation DT(P) such that no point in P is inside the circumcircle o ...

de Longchamps point In geometry, the de Longchamps point of a triangle is a triangle center named after French mathematician Gaston Albert Gohierre de Longchamps. It is the reflection of the orthocenter of the triangle about the circumcenter.. Definition Let the ...

Desargues' theorem In projective geometry, Desargues's theorem, named after Girard Desargues, states: :Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''. Denote the three vertices of one triangle by and , and tho ...

Droz-Farny line theorem In Euclidean geometry, the Droz-Farny line theorem is a property of two perpendicular lines through the orthocenter of an arbitrary triangle. Let T be a triangle with vertices A, B, and C, and let H be its orthocenter (the common point of its thr ...

Encyclopedia of Triangle Centers The Encyclopedia of Triangle Centers (ETC) is an online list of thousands of points or "centers" associated with the geometry of a triangle. It is maintained by Clark Kimberling, Professor of Mathematics at the University of Evansville. , the l ...

Equal incircles theorem In geometry, the equal incircles theorem derives from a Japanese Sangaku, and pertains to the following construction: a series of rays are drawn from a given point to a given line such that the inscribed circles of the triangles formed by adjacent ...

* Equal parallelians point *

Equidissection In geometry, an equidissection is a partition of a polygon into triangles of equal area. The study of equidissections began in the late 1960s with Monsky's theorem, which states that a square cannot be equidissected into an odd number of triang ...

Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

Euler's line In geometry, the Euler line, named after Leonhard Euler (), is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, inclu ...

Euler's theorem in geometry In geometry, Euler's theorem states that the distance ''d'' between the circumcenter and incenter of a triangle is given by d^2=R (R-2r) or equivalently \frac + \frac = \frac, where R and r denote the circumradius and inradius respectively (the ...

Erdős–Mordell inequality In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ''ABC'' and point ''P'' inside ''ABC'', the sum of the distances from ''P'' to the sides is less than or equal to half of the sum of the distances from ''P'' to the ...

Exeter point In geometry, the Exeter point is a special point associated with a plane triangle. The Exeter point is a triangle center and is designated as the center X(22) in Clark Kimberling's Encyclopedia of Triangle Centers. This was discovered in a comput ...

Exterior angle theorem The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geo ...

Fagnano's problem In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775: The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle. Solution The orthic tria ...

Fermat point In Euclidean geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat–Torricelli point, is a point such that the sum of the three distances from each of the three vertices of the triangle to the point is the smallest ...

Fermat's right triangle theorem Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat. It has several equivalent formulations, one o ...

* Fuhrmann circle * Fuhrmann triangle *

Geometric mean theorem The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states ...

* GEOS circle *

Gergonne point In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

Golden triangle (mathematics) A golden triangle, also called a sublime triangle, is an isosceles triangle in which the duplicated side is in the golden ratio \varphi to the base side: : = \varphi = \approx 1.618~034~. Angles * The vertex angle is: ::\theta = 2\arcsin = 2\ar ...

* Gossard perspector * Hadley's theorem *

Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ ...

Heilbronn triangle problem In discrete geometry and discrepancy theory, the Heilbronn triangle problem is a problem of placing points in the plane, avoiding triangles of small area. It is named after Hans Heilbronn, who conjectured that, no matter how points are placed ...

Heptagonal triangle A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter an ...

Heronian triangle In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths , , and and area are all integers. Heronian triangles are named after Heron of Alexandria, based on their relation to Heron's formula. Heron's formula implies ...

Heron's formula In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . If s = \tfrac12(a + b + c) is the semiperimeter of the triangle, the area is, :A = \sqrt. It is named after first-century ...

Hofstadter points In triangle geometry, a Hofstadter point is a special point associated with every plane triangle. In fact there are several Hofstadter points associated with a triangle. All of them are triangle centers. Two of them, the Hofstadter zero-point an ...

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 points called ''angles'' or ''vertices''. Just as in the Euclidean case, three po ...

(non-Euclidean geometry) *

Hypotenuse In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...

Incircle and excircles of a triangle In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. ...

Inellipse In triangle geometry, an inellipse is an ellipse that touches the three sides of a triangle. The simplest example is the incircle. Further important inellipses are the Steiner inellipse, which touches the triangle at the midpoints of its sides, t ...

Integer triangle An integer triangle or integral triangle is a triangle all of whose sides have lengths that are integers. A rational triangle can be defined as one having all sides with rational Rationality is the quality of being guided by or based on re ...

Isodynamic point In Euclidean geometry, the isodynamic points of a triangle are points associated with the triangle, with the properties that an inversion centered at one of these points transforms the given triangle into an equilateral triangle, and that the dis ...

Isogonal conjugate __notoc__ In geometry, the isogonal conjugate of a point with respect to a triangle is constructed by reflecting the lines about the angle bisectors of respectively. These three reflected lines concur at the isogonal conjugate of . (This ...

Isoperimetric point In geometry, the isoperimetric point is a special point associated with a plane triangle. The term was originally introduced by G.R. Veldkamp in a paper published in the American Mathematical Monthly in 1985 to denote a point ''P'' in the plane of ...

Isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

Isosceles triangle theorem In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Eu ...

Isotomic conjugate In geometry, the isotomic conjugate of a point with respect to a triangle is another point, defined in a specific way from and : If the base points of the lines on the sides opposite are reflected about the midpoints of their respective sid ...

* Isotomic lines *

Jacobi point In plane geometry, a Jacobi point is a point in the Euclidean plane determined by a triangle ''ABC'' and a triple of angles ''α'', ''β'', and ''γ''. This information is sufficient to determine three points ''X'', ''Y'', and ''Z'' ...

Japanese theorem for concyclic polygons __notoc__ In geometry, the Japanese theorem states that no matter how one triangulates a cyclic polygon, the sum of inradii of triangles is constant.Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Conversel ...

Johnson circles In geometry, a set of Johnson circles comprises three circles of equal radius sharing one common point of intersection . In such a configuration the circles usually have a total of four intersections (points where at least two of them meet): th ...

Kepler triangle A Kepler triangle is a special right triangle with edge lengths in geometric progression. The ratio of the progression is \sqrt\varphi where \varphi=(1+\sqrt)/2 is the golden ratio, and the progression can be written: or approximately . Squares ...

Kobon triangle problem The Kobon triangle problem is an unsolved problem in combinatorial geometry first stated by Kobon Fujimura (1903-1983). The problem asks for the largest number ''N''(''k'') of nonoverlapping triangles whose sides lie on an arrangement of ''k'' ...

Kosnita's theorem In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle. Let ABC be an arbitrary triangle, O its circumcenter and O_a,O_b,O_c are the circumcenters of three triangles OBC, OCA, and OAB resp ...

Leg (geometry) In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a "side about the right angle". The side opposit ...

Lemoine's problem In mathematics, Lemoine's problem is a certain construction problem in elementary plane geometry posed by the French mathematician Émile Lemoine (1840–1912) in 1868. The problem was published as Question 864 in '' Nouvelles Annales de Math� ...

Lester's theorem In Euclidean plane geometry, Lester's theorem states that in any scalene triangle, the two Fermat points, the nine-point center, and the circumcenter lie on the same circle. The result is named after June Lester, who published it in 1997, and t ...

List of triangle inequalities In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle, or for every triangle meeting certain conditions. The inequalities give an ordering of two different values: they are of the ...

Mandart inellipse In geometry, the Mandart inellipse of a triangle is an ellipse inscribed within the triangle, tangent to its sides at the contact points of its excircles (which are also the vertices of the extouch triangle and the endpoints of the splitters). ...

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 Stat ...

Medial triangle In Euclidean geometry, the medial triangle or midpoint triangle of a triangle is the triangle with vertices at the midpoints of the triangle's sides . It is the case of the midpoint polygon of a polygon with sides. The medial triangle is not ...

Median (geometry) In geometry, a median of a triangle is a line segment joining a vertex (geometry), vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each ...

Menelaus' theorem Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ''ABC'', and a transversal line that crosses ''BC'', ''AC'', and ''AB'' at points ''D'', ''E'', and ''F'' respec ...

Miquel's theorem Miquel's theorem is a result in geometry, named after Auguste Miquel, concerning the intersection of three circles, each drawn through one vertex of a triangle and two points on its adjacent sides. It is one of several results concerning circles ...

Mittenpunkt In geometry, the (from German: ''middle point'') of a triangle is a triangle center: a point defined from the triangle that is invariant under Euclidean transformations of the triangle. It was identified in 1836 by Christian Heinrich von Nagel ...

Modern triangle geometry In mathematics, modern triangle geometry, or new triangle geometry, is the body of knowledge relating to the properties of a triangle discovered and developed roughly since the beginning of the last quarter of the nineteenth century. Triangles and ...

Monsky's theorem In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection. The problem was posed by Fred Richman in the '' American ...

Morley centers In geometry the Morley centers are two special points associated with a plane triangle. Both of them are triangle centers. One of them called first Morley center (or simply, the Morley center ) is designated as X(356) in Clark Kimberling's Encyclo ...

Morley triangle In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem ...

Morley's trisector theorem In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle. The theorem ...

* Musselman's theorem *

Nagel point In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concur ...

Napoleon points In geometry, Napoleon points are a pair of special points associated with a plane triangle. It is generally believed that the existence of these points was discovered by Napoleon Bonaparte, the Emperor of the French from 1804 to 1815, but many have ...

Napoleon's theorem In geometry, Napoleon's theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward or all inward, the lines connecting the centres of those equilateral triangles themselves form an equilateral tr ...

Nine-point circle In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: * The midpoint of eac ...

Nine-point hyperbola In Euclidean geometry with triangle , the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic: :Given ...

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-thi ...

Orthocenter In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to (i.e., forming a right angle with) a line containing the base (the side opposite the vertex). This line containing the opposite side is called the '' ...

Orthocentric system In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and t ...

Orthocentroidal circle In geometry, the orthocentroidal circle of a non-equilateral triangle is the circle that has the triangle's orthocenter and centroid at opposite ends of its diameter. This diameter also contains the triangle's nine-point center and is a subset o ...

* Orthopole * Pappus' area theorem * Parry point *

Pedal triangle In geometry, a pedal triangle is obtained by projecting a point onto the sides of a triangle. More specifically, consider a triangle ''ABC'', and a point ''P'' that is not one of the vertices ''A, B, C''. Drop perpendiculars from ''P'' to the thr ...

* 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 : \begin r^2 & = HA\times HD=HB\times HE=HC\times HF \\ & =-4R^2\cos A \cos B \cos C=4R^2-\frac(a^2+b^2+c^2), \end ...

Pompeiu's theorem Pompeiu's theorem is a result of plane geometry, discovered by the Romanian mathematician Dimitrie Pompeiu. The theorem is simple, but not classical. It states the following: :''Given an equilateral triangle ABC in the plane, and a point P in t ...

Pons asinorum In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the ''pons asinorum'' (, ), typically translated as "bridge of asses". This statement is Proposition 5 of Book 1 in Eu ...

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 In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows: :Let ''A'', ''B'' be the endpoints of the hypotenuse of a right triangle ''ABC''. Let ''D'' be t ...

Reuleaux triangle A Reuleaux triangle is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed from the intersection of three circular disks, each having its center on the boundary of the ...

Regiomontanus Johannes Müller von Königsberg (6 June 1436 – 6 July 1476), better known as Regiomontanus (), was a mathematician, astrologer and astronomer of the German Renaissance, active in Vienna, Buda and Nuremberg. His contributions were instrumental ...

* Regiomontanus' angle maximization problem * Reuschle's theorem *

Right triangle A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...

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 A ...

Scalene triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...

Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere (spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be defined mor ...

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

Similarity (geometry) In Euclidean geometry, two objects are similar if they have the same shape, or 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 red ...

* Similarity system of triangles *

Simson line In geometry, given a triangle and a point on its circumcircle, the three closest points to on lines , , and are collinear. The line through these points is the Simson line of , named for Robert Simson. The concept was first published, however ...

Special right triangles A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45° ...

Spieker center In geometry, the Spieker center is a special point associated with a plane triangle. It is defined as the center of mass of the perimeter of the triangle. The Spieker center of a triangle is the center of gravity of a homogeneous wire frame in t ...

Spieker circle In geometry, the incircle of the medial triangle of a triangle is the Spieker circle, named after 19th-century German geometer Theodor Spieker. Its center, the Spieker center, in addition to being the incenter of the medial triangle, is the cen ...

Spiral of Theodorus In geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein 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 ...

Splitter (geometry) In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle. They are not to be confused with cleavers, which also bisect the perimeter but instea ...

Steiner circumellipse In geometry, the Steiner ellipse of a triangle, also called the Steiner circumellipse to distinguish it from the Steiner inellipse, is the unique circumellipse ( ellipse that touches the triangle at its vertices) whose center is the triangle's ...

Steiner inellipse In geometry, the Steiner inellipse,Weisstein, E. "Steiner Inellipse" — From MathWorld, A Wolfram Web Resource, http://mathworld.wolfram.com/SteinerInellipse.html. midpoint inellipse, or midpoint ellipse of a triangle is the unique ellipse insc ...

* Steiner–Lehmus theorem *

Stewart's theorem In geometry, Stewart's theorem yields a relation between the lengths of the sides and the length of a cevian in a triangle. Its name is in honour of the Scottish mathematician Matthew Stewart, who published the theorem in 1746. Statement Let ...

* Steiner point *

Symmedian In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corr ...

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 vertices. Thus the incircle of the ...

Tarry point In geometry, the Tarry point for a triangle is a point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangle's first Brocard triangle . The Tarry point lies on the other end ...

Ternary plot A ternary plot, ternary graph, triangle plot, simplex plot, Gibbs triangle or de Finetti diagram is a barycentric plot on three variables which sum to a constant. It graphically depicts the ratios of the three variables as positions in an equi ...

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

Thomson cubic In geometry, the Thomson cubic of a triangle is the locus of centers of circumconics whose normals at the vertices are concurrent Concurrent means happening at the same time. Concurrency, concurrent, or concurrence may refer to: Law * Concur ...

Triangle center In geometry, a triangle center (or triangle centre) is a point in the plane that is in some sense a center of a triangle akin to the centers of squares and circles, that is, a point that is in the middle of the figure by some measure. For example ...

Triangle conic In triangle geometry, a triangle conic is a conic in the plane of the reference triangle and associated with it in some way. For example, the circumcircle and the incircle of the reference triangle are triangle conics. Other examples are the Steine ...

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 triangle ...

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 degenerate triangles, but ...

Triangular bipyramid In geometry, the triangular bipyramid (or dipyramid) is a type of hexahedron, being the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces. As the name suggests, i ...

Triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...

Triangular pyramid In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

Triangular tiling In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

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 me ...

Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is t ...

Trilinear polarity In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the Plane (geometry), plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the Vertex (g ...

Trisected perimeter point In geometry, given a triangle ''ABC'', there exist unique points ''A´'', ''B´'', and ''C´'' on the sides ''BC'', ''CA'', ''AB'' respectively, such that: :* ''A´'', ''B´'', and ''C´'' partition the perimeter of the triangle into three equal- ...

Viviani's theorem Viviani's theorem, named after Vincenzo Viviani, states that the sum of the 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 various ma ...

* Wernau points *

Yff center of congruence In geometry, the Yff center of congruence is a special point associated with a triangle. This special point is a triangle center and Peter Yff initiated the study of this triangle center in 1987. Isoscelizer An isoscelizer of an angle ''A'' in ...

Trigonometry

Differentiation of trigonometric functions The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the derivative of the sine function is written sin′ ...

Exact trigonometric constants In mathematics, the values of the trigonometric functions can be expressed approximately, as in \cos (\pi/4) \approx 0.707, or exactly, as in \cos (\pi/ 4)= \sqrt 2 /2. While trigonometric tables contain many approximate values, the exact values f ...

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

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

Law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

Law of cotangents In trigonometry, the law of cotangentsThe Universal Encyclopaedia of Mathematics, Pan Reference Books, 1976, page 530. English version George Allen and Unwin, 1964. Translated from the German version Meyers Rechenduden, 1960. is a relationship am ...

Law of sines In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, \frac \,=\, \frac \,=\, \frac \,=\, 2R, where , and a ...

Law of tangents In trigonometry, the law of tangents is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, , , and are the lengths of the three sides of the triangle, and , , ...

List of integrals of inverse trigonometric functions The following is a list of indefinite integrals (antiderivatives) of expressions involving the inverse trigonometric functions. For a complete list of integral formulas, see lists of integrals. * The inverse trigonometric functions are also known ...

List of integrals of trigonometric functions The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antider ...

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 In trigonometry, Mollweide's formula is a pair of relationships between sides and angles in a triangle. A variant in more geometrical style was first published by Isaac Newton in 1707 and then by in 1746. Thomas Simpson published the now-standar ...

* Outline of trigonometry *

Rational trigonometry ''Divine Proportions: Rational Trigonometry to Universal Geometry'' is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called rational trigonometry. The book advocat ...

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 that is oppo ...

Solution of triangles Solution of triangles ( la, solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths of sides), when some of these are known. The triangle can be located on a plane or on a sphere. Appl ...

Spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

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 In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expressions. In calculus, trigonometric substitution is a technique for evaluating integrals. Moreover, one may use the trigonometric identities t ...

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 w ...

Trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

Uses of trigonometry Amongst the lay public of non-mathematicians and non-scientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the music theory, theory of ...

Applied mathematics

De Finetti diagram A de Finetti diagram is a ternary plot used in population genetics. It is named after the Italian statistician Bruno de Finetti (1906–1985) and is used to graph the genotype frequencies of populations, where there are two alleles and the popul ...

Triangle mesh In computer graphics, a triangle mesh is a type of polygon mesh. It comprises a set of triangles (typically in three dimensions) that are connected by their common edges or vertices. Many graphics software packages and hardware devices can ...

Nonobtuse mesh In computer graphics, a nonobtuse triangle mesh is a polygon mesh composed of a set of triangles in which no angle is obtuse, ''i.e.'' greater than 90°. If each (triangle) face angle is strictly less than 90°, then the triangle mesh is said to b ...

Resources

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Pythagorean Triangles ''Pythagorean Triangles'' is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language by Wacław Sierpiński (titled ''Trójkąty pitagorejskie''), and published in Warsaw in 1 ...

'' * ''

The Secrets of Triangles ''The Secrets of Triangles: A Mathematical Journey'' is a popular mathematics book on the geometry of triangles. It was written by Alfred S. Posamentier and , and published in 2012 by Prometheus Books. Topics The book consists of ten chapters, ...

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 ...

(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...

Number theory

Triangular arrays of numbers

Bell numbers In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan. In an example of Stigler's law of eponymy ...

Boustrophedon transform In mathematics, the boustrophedon transform is a procedure which maps one sequence to another. The transformed sequence is computed by an "addition" operation, implemented as if filling a triangular array in a boustrophedon (zigzag- or serpentine-l ...

Eulerian number In combinatorics, the Eulerian number ''A''(''n'', ''m'') is the number of permutations of the numbers 1 to ''n'' in which exactly ''m'' elements are greater than the previous element (permutations with ''m'' "ascents"). They are the coefficients ...

Floyd's triangle Floyd's triangle is a triangular array of natural numbers used in computer science education. It is named after Robert Floyd. It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1 in the top left corner: T ...

* 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 counting problems. They are named after Canadian mathematic ...

Rencontres numbers In combinatorial mathematics, the rencontres numbers are a triangular array of integers that enumerate permutations of the set with specified numbers of fixed points: in other words, partial derangements. (''Rencontre'' is French for ''encounter' ...

Romberg's method In numerical analysis, Romberg's method is used to estimate the 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 a triangul ...

Stirling numbers of the first kind In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed poin ...

Stirling numbers of the second kind In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...

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 A tetrahedral number, or triangular pyramidal number, is a figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The th tetrahedral number, , is the sum of the first triangular numbers, that is, ...

The (incomplete)

Bell polynomials In combinatorial mathematics, the Bell polynomials, named in honor of Eric Temple Bell, are used in the study of set partitions. They are related to Stirling and Bell numbers. They also occur in many applications, such as in the Faà di Bruno's fo ...

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 en ...

Integers in triangle geometry

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

Eisenstein triple Similar to a Pythagorean triple, an Eisenstein triple (named after Gotthold Eisenstein) is a set of integers which are the lengths of the sides of a triangle where one of the angles is 60 or 120 degrees. The relation of such triangles to the Eisens ...

Geography

Bermuda Triangle The Bermuda Triangle, also known as the Devil's Triangle, is an urban legend focused on a loosely defined region in the western part of the North Atlantic Ocean where a number of aircraft and ships are said to have disappeared under mysterio ...

Historic Triangle The Historic Triangle includes three historic colonial communities located on the Virginia Peninsula of the United States and is bounded by the York River on the north and the James River on the south. The points that form the triangle are James ...

Lithium Triangle The Lithium Triangle ( es, Triángulo del Litio) is a region of the Andes rich in lithium reserves around the borders of Argentina, Bolivia, and Chile. The lithium in the triangle is concentrated in various salt pans that exist along the Atac ...

Parliamentary Triangle, Canberra The National Triangle, which is referred to as the Parliamentary Triangle, is the ceremonial precinct of Canberra, containing some of Australia's most significant buildings. The National Triangle is formed by Commonwealth, Kings and Constituti ...

Research Triangle The Research Triangle, or simply The Triangle, are both common nicknames for a metropolitan area in the Piedmont region of North Carolina in the United States, anchored by the cities of Raleigh and Durham and the town of Chapel Hill, home to ...

Sunni Triangle The Sunni Triangle is a densely populated region of Iraq to the north and west of Baghdad inhabited mostly by Sunni Muslim Arabs. The roughly triangular area's points are usually said to lie near Baghdad (the southeast point), Ramadi (the southwest ...

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 t ...

Anatomy

Submandibular triangle The submandibular triangle (or submaxillary or digastric triangle) corresponds to the region of the neck immediately beneath the body of the mandible. Boundaries and coverings It is bounded: * ''above'', by the lower border of the body of the man ...

* Triangle choke *

Arm triangle choke Arm triangle choke, side choke, or head and arm choke are generic terms describing blood chokeholds in which the opponent is strangled in between their own shoulder and the practitioner's arm. This is as opposed to the regular triangle choke, whi ...

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 The carotid triangle (or superior carotid triangle) is a portion of the anterior triangle of the neck. Coverings and boundaries It is bounded: * Posteriorly by the anterior border of the Sternocleidomastoid; * Anteroinferiorly, by the superior be ...

Clavipectoral triangle The clavipectoral triangle (also known as the deltopectoral triangle) is an anatomical region found in humans and other animals. It is bordered by the following structures: * Clavicle (superiorly) * Lateral border of Pectoralis Major (medially) ...

Inguinal triangle In human anatomy, the inguinal triangle is a region of the abdominal wall. It is also known by the eponym Hesselbach's triangle, after Franz Kaspar Hesselbach. Structure It is defined by the following structures: * Medial border: Lateral margi ...

Codman triangle The Codman triangle (previously referred to as Codman's triangle) is the triangular area of new subperiosteal bone that is created when a lesion, often a tumour, raises the periosteum away from the bone. A Codman triangle is not actually a full t ...

Artifacts

Black triangle Black triangle may refer to one of the following: * Black triangle (badge), a Nazi concentration camp badge worn by inmates deemed "asocial" ** LGBT symbols#Triangle badges of Nazi Germany, Lesbian or feminist symbol reclaimed from the Nazi use * A ...

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

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 dif ...

Triquetra The triquetra ( ; from the Latin adjective ''triquetrus'' "three-cornered") is a triangular figure composed of three interlaced arcs, or (equivalently) three overlapping '' vesicae piscis'' lens shapes. It is used as an ornamental design in ar ...

Symbols

Eye of Providence The Eye of Providence (or the All-Seeing Eye of God) is a symbol that depicts an eye, often enclosed in a triangle and surrounded by Ray (optics), rays of light or Glory (optical phenomenon), glory, meant to represent divine providence, whereby ...

Valknut The valknut is a symbol consisting of three interlocked triangles. It appears on a variety of objects from the archaeological record of the ancient Germanic peoples. The term ''valknut'' is a modern development; it is not known what term or term ...

Shield of the Trinity The Shield of the Trinity or ''Scutum Fidei'' (Latin for "shield of faith") is a traditional Christian visual symbol which expresses many aspects of the doctrine of the Trinity, summarizing the first part of the Athanasian Creed in a compact diag ...

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