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Problem of finding unknown lengths and angles of a triangle
Solution of triangles (Latin: solutio triangulorum) is the main trigonometric problem of finding the characteristics of a triangle (angles and lengths
Solution_of_triangles
Geometry of figures on the surface of a sphere
be restricted to spherical triangles, referred to simply as triangles. Both vertices and angles at the vertices of a triangle are denoted by the same upper
Spherical_trigonometry
Generalization of Pythagorean theorem
are given. The theorem is used in solution of triangles, i.e., to find (see Figure 3): the third side of a triangle if two sides and the angle between
Law_of_cosines
Shape with three equal sides
asked whether a triangle can be constructed by the vertex of equilateral triangles attached on the sides of a triangle. The solution was provided by German
Equilateral_triangle
Overview of and topical guide to trigonometry
Proofs of trigonometric identities Pythagorean trigonometric identity Tangent half-angle formula Solution of triangles Law of sines Law of cosines Law of tangents
Outline_of_trigonometry
Similar triangles that share two side lengths
triangles are said to be 5-Con or almost congruent if they are not congruent triangles but they are similar triangles and share two side lengths (of non-corresponding
5-Con_triangles
Topics referred to by the same term
journal of semiotics Super soft source or supersoft X-ray source sss, the quark content of an omega baryon (Ω−) Side-side-side, solution of triangles with
SSS
Topics referred to by the same term
Congruence of triangles may refer to: Congruence (geometry)#Congruence of triangles Solution of triangles This disambiguation page lists articles associated
Congruence_of_triangles
Property of all triangles on a Euclidean plane
spherical triangles Law of cosines Law of tangents Law of cotangents Mollweide's formula – for checking solutions of triangles Solution of triangles Surveying
Law_of_sines
Unsolved problem in combinatorial geometry
lines, 5 triangles 6 lines, 7 triangles 7 lines, 11 triangles Roberts's triangle theorem, on the minimum number of triangles that n {\displaystyle n} lines
Kobon_triangle_problem
Triangle whose side lengths and area are integers
rational solutions of the above equation) are sometimes also called Heronian triangles or rational triangles; in this article, these more general triangles will
Heronian_triangle
Mathematical relation in spherical triangles
Regiomontanus in 1537. Half-side formula Hyperbolic law of cosines Solution of triangles Spherical law of sines W. Gellert; H. Küstner; M. Hellwich; H. Kästner
Spherical_law_of_cosines
Early study of triangles can be traced to Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics during the 2nd millennium BC. Systematic
History_of_trigonometry
Problem of minimizing sum of transport costs
respect to triangles △ACD, △BCD; This leads to draw two other equilateral triangles △ACF, △BCG, where F, G are located outside the △ABC triangle, as well
Weber_problem
Type of triangle
trigonometry, a skinny triangle[citation needed] is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified
Skinny_triangle
Optical illusion
right-angled triangle, but one has a 1×1 hole in it. The key to the puzzle is the fact that neither of the 13×5 "triangles" is truly a triangle, nor would
Missing_square_puzzle
Triangle containing a 90-degree angle
obtuse triangles (oblique triangles) Spiral of Theodorus Trirectangular spherical triangle Artmann, Benno (2012) [1999], Euclid: The Creation of Mathematics
Right_triangle
Uses of trigonometry De Finetti diagram Triangle mesh Nonobtuse mesh Encyclopedia of Triangle Centers Pythagorean Triangles The Secrets of Triangles Triangular
List_of_triangle_topics
Triangle with specific characteristics
Brahmagupta triangles is an old problem. A closed form solution of the problem was found by Reinhold Hoppe in 1880. Let the side lengths of a Brahmagupta
Brahmagupta_triangle
On point sets with no small-area triangles
Heilbronn triangle problem asks how to place n {\displaystyle n} points in a region of the plane, such as a square, so that the smallest of the triangles formed
Heilbronn_triangle_problem
Theorem on an equilateral triangle constructed from three equilateral triangles
equilateral triangles constructed on its sides' exteriors, and points L, M, N are the centroids of those triangles. The theorem for outer triangles states
Napoleon's_theorem
German astronomer and mathematician (1561–1613)
(Trigonometry: A short and clear treatise on the solution of triangles) which was printed as an appendix to work of Abraham Scultetus Sphæricorum libri tres methodice
Bartholomaeus_Pitiscus
Triangle with circular arc edges
triangles with the same interior angles as each other are equivalent to each other under Möbius transformations. Circular triangles give the solution
Circular_triangle
Relation between sides of a right triangle
right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square
Pythagorean_theorem
Triangle with integer side lengths
only such triangles are rational-sided equilateral triangles. Any triple of positive integers can serve as the side lengths of an integer triangle as long
Integer_triangle
Tiling puzzle
triangular grid made of equilateral triangles. Its sides alternate in length: six sides coincide with the grid and are 7 triangles (placed edge-to-edge)
Eternity_puzzle
Optimisation problem in triangle geometry
triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof
Fagnano's_problem
Spherical triangle that can be used to tile a sphere
called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while
Schwarz_triangle
combining two triangles to form a quadrilateral, then splitting the quadrilateral in the other direction to produce two new triangles. Flipping is used
Types_of_mesh
Triangle with at least two sides congruent
mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as
Isosceles_triangle
Problem in trigonometry
original contribution, but merely restated Snellius 75 years later. Solution of triangles Triangulation (surveying) Bowser: A treatise Norman J. Wildberger
Snellius–Pothenot_problem
Limiting case which is different from the rest of the class
special cases are degenerate. For example, right triangles, isosceles triangles and equilateral triangles are non-generic and non-degenerate. In fact, degenerate
Degeneracy_(mathematics)
Rational right triangles cannot have square area
right triangle and a square with equal areas cannot have all sides commensurate with each other. There do not exist two integer-sided right triangles in
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Right triangle with a feature making calculations on the triangle easier
special right triangles are those involving some special relationship between the triangle's three angle measures. The angles of these triangles are such that
Special_right_triangle
Topics referred to by the same term
Side-side-side is a means of analyzing triangles discussed at: Solution of triangles § Three sides given (SSS) SSS postulate This disambiguation page lists
Side-side-side
Two tetrahedra joined by one face
which the apices of both pyramids are on a line passing through the center of the base, such that its faces are isosceles triangles. If two tetrahedra
Triangular_bipyramid
Integer side lengths of a right triangle
Heron triangles which cannot be decomposed into two integer right triangles (PDF), 41st Meeting of Florida Section of Mathematical Association of America
Pythagorean_triple
Problem solving strategy
then the full solution can follow by extension. For example, to calculate the area of a regular hexagon, divide it into equilateral triangles, calculate
Symmetry_in_problem_solving
One can't dissect a square into an odd number of triangles of equal area
a dissection into n triangles of equal area, then the area of each triangle is 1/n. Colour each point in the square with one of three colours, depending
Monsky's_theorem
Unsolved problem about sums of powers
Example-1: (5,12,13) and (6,8,10) are the appropriate Pythagorean Triangles for the solution. 5 + 12 - 13 = 6 + 8 - 10 = 4 If you apply to the following equations
Prouhet–Tarry–Escott_problem
Qualification in mathematics study
Functions, Systems of Linear Equalities, Indices, Surds, Logaritms, Progressions, Linear Law, Coordinate Geometry, Vectors, Solution of Triangles, Index Numbers
Additional_Mathematics
Geometric construction problem relating to triangles
1868: Given one vertex of each of the equilateral triangles placed on the sides of a triangle, construct the original triangle. The problem was published
Lemoine's_problem
On triangles in line arrangements
theorem guarantees that three triangles will exist, but the solution to the Kobon triangle problem has five triangles. Despite the fact that these seem
Roberts's_triangle_theorem
Intersection of triangle altitudes
three. These four possible triangles will all have the same nine-point circle. Consequently these four possible triangles must all have circumcircles
Orthocenter
Line constructed from a triangle
isosceles triangles, for which the Euler line coincides with the symmetry axis of the triangle and contains all triangle centers. The tangential triangle of a
Euler_line
Trigonometric relation between sides and angles of a triangle
of solutions of triangles. Let a , {\displaystyle a,} b , {\displaystyle b,} and c {\displaystyle c} be the lengths of the three sides of a triangle.
Mollweide's_formula
Triangulation method
the set are outside of it. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles. The triangulation
Delaunay_triangulation
of sine and cosine expressed in surds) Exsecant Half-side formula Hyperbolic function Laws for solution of triangles: Law of cosines Spherical law of
List of trigonometric identities
List_of_trigonometric_identities
Dynamical system involving reflection
paths were stable, all triangles would be covered by the stable neighborhoods of rational triangles. However, there exist triangles for which all periodic
Triangular_billiards
Topics referred to by the same term
The concept of included angle is discussed at: Congruence of triangles Solution of triangles This disambiguation page lists mathematics articles associated
Included_angle
Arrangement of numbers
than the hexagon of hexagons. As with the above, the rows of triangles run in three directions and there are 24 triangles in a T-hexagon of order 2. In general
Magic_hexagon
Method for transmitting packets of data
Triangular routing is a method for transmitting packets of data in communications networks. It uses a form of routing that sends a packet to a proxy system before
Triangular_routing
Mathematical puzzle game
distributions of disks and the edges representing moves. For one disk, the graph is a triangle: The graph for two disks is three triangles connected to
Tower_of_Hanoi
Geometric problems involving the partition of a figure
in three dimension and any two zonohedra of equal volume (in any dimension). A partition into triangles of equal area is called an equidissection. Most
Dissection_problem
Four-dimensional analogue of the tetrahedron
vertices of the tetrahedron. This cannot be done in 3-dimensional space. The regular 5-cell is a solution to the problem: Make 10 equilateral triangles, all
5-cell
Square whose vertices lie on a triangle
known to have a solution for every polygon and for every convex set, two special cases that both apply to triangles. Every acute triangle has three inscribed
Inscribed square in a triangle
Inscribed_square_in_a_triangle
Curved triangle with constant width
and the area is that of the Reuleaux triangle, the Reuleaux triangle is the optimal enclosure. Circular triangles are triangles with circular-arc edges
Reuleaux_triangle
Number, approximately 2.41421
pair of grey triangles on the sides has perpendicular diagonals in ratio σ {\displaystyle \sigma } , hence is a silver rhombus. If the triangles have
Silver_ratio
Subdivision of a planar object into triangles
subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional
Triangulation_(geometry)
Geometric construction of a smaller triangle
suggestion of Hugo Steinhaus is that the (central) triangle with sides p,q,r be reflected in its sides and vertices. These six extra triangles partially
One-seventh_area_triangle
Region of Texas that contains the state's five largest cities
Zhang. "REINVENTING THE TEXAS TRIANGLE Solutions for Growing Challenges" (PDF). The University of Texas at Austin School of Architecture Center for Sustainable
Texas_Triangle
the triangles of the graph. No six points can include three triangles without either two of the three triangles sharing an edge or all three triangles forming
Ruzsa–Szemerédi_problem
Triangle center minimizing sum of distances to each vertex
no angle of the triangle exceeds 120°. Fig. 2 shows the equilateral triangles △ARB, △AQC, △CPB attached to the sides of the arbitrary triangle △ABC. Here
Fermat_point
point set or polygon) into triangles, meeting edge-to-edge. Both input points and Steiner points may be used as triangle vertices. Delaunay refinement
Steiner point (computational geometry)
Steiner_point_(computational_geometry)
Perpendicular line segment from a triangle's side to opposite vertex
triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an obtuse angle), the foot of the
Altitude_(triangle)
Geometry puzzle
repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles
Langley's_Adventitious_Angles
Isogonal polyhedron with regular faces
colored triangles on a sphere: There are 16 fundamental triangles, visible in the faces of the octagonal bipyramid and alternately colored triangles on a
Uniform_polyhedron
Conic curves associated with a triangle
Lemoine in 1868: "Construct a triangle, given the peaks of the equilateral triangles constructed on the sides." A solution to the problem was published
Kiepert_conics
fundamental triangle, (p q r), defined by internal angles as π/p, π/q, and π/r. Special cases are right triangles (p q 2). Uniform solutions are constructed
Lists of uniform tilings on the sphere, plane, and hyperbolic plane
Lists_of_uniform_tilings_on_the_sphere,_plane,_and_hyperbolic_plane
Matrix used in finite element analysis
for the numerical solution of elliptic partial differential equations, the stiffness matrix is a matrix that represents the system of linear equations
Stiffness_matrix
Convex polyhedron with 16 triangular faces
resulting polyhedron has 16 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces is called a deltahedron. There are
Gyroelongated square bipyramid
Gyroelongated_square_bipyramid
French mathematician (d. 1674)
History of the theory of numbers. Vol. II: Diophantine analysis. pp. 184–186. Pepin T (1880). "Solution d'un Problème de Frenicle Sur Deux Triangles Rectangles"
Bernard_Frénicle_de_Bessy
Convex polyhedron with 14 triangle faces
equilateral triangles, so that the resulting polyhedron has 14 equilateral triangles as its faces. A polyhedron with only equilateral triangles as faces
Triaugmented_triangular_prism
Convex polyhedron with 12 triangular faces
of such constructions, the snub disphenoid has 12 equilateral triangles. A deltahedron is a polyhedron in which all faces are equilateral triangles.
Snub_disphenoid
Polynomial equation of degree 3
is an equation of the form a x 3 + b x 2 + c x + d = 0 {\displaystyle ax^{3}+bx^{2}+cx+d=0} in which a is not zero. The solutions of this equation are
Cubic_equation
Two pentagonal pyramids fused base-to-base
one of the eight convex deltahedra if the faces of two pyramids are equilateral triangles and all edges are of equal length. It is an example of a composite
Pentagonal_bipyramid
Type of metric geometry
guarantees triangle congruence. Take, for example, two right isosceles taxicab triangles whose angles measure 45-90-45. The two legs of both triangles have
Taxicab_geometry
Three tangent circles in a triangle
different triangles. Bottema (2001) credits the enumeration of these solutions to Pampuch (1904), but Cajori (1893) notes that this count of the number of solutions
Malfatti_circles
Area of a right triangle with rational-numbered sides
1201/70)} . Both of these right triangles have area n = 6 {\displaystyle n=6} . According to Leonard Eugene Dickson's compilation of historical texts
Congruent_number
Polynomial equation whose integer solutions are sought
coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates the sum of two or more unknowns, with coefficients
Diophantine_equation
Regular 5-polytope
The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick
5-simplex
Number, approximately 1.618
central golden triangle. The five points of a regular pentagram are golden triangles, as are the ten triangles formed by connecting the vertices of a regular
Golden_ratio
Physics problem related to laws of motion and gravity
Newton's law of universal gravitation. Unlike the two-body problem, the three-body problem has no general closed-form analytic solution. The differential
Three-body_problem
Cuboid whose edges and face diagonals have integer lengths
following Heronian triangles exist: A Heronian triangle with side lengths ( d 2 , e 2 , f 2 ) {\displaystyle (d^{2},e^{2},f^{2})} , an area of a b c g {\displaystyle
Euler_brick
Mathematical puzzle
involve Pythagorean triples for the two right triangles with sides (A, w, b) and (B, w, a) and integer solutions of the optic equation 1 A + 1 B = 1 h . {\displaystyle
Crossed_ladders_problem
Geometry puzzle
number of polyform (polyomino, polyiamond, and polyhex) solutions, with no holes. 11 squares 12 squares 10 triangles 22 triangles 26 triangles 4 hexagons
Tetrad_(geometry_puzzle)
Probability theory paradox
method of random selection is specified, the problem will have a well-defined solution (determined by the principle of indifference). The three solutions presented
Bertrand paradox (probability)
Bertrand_paradox_(probability)
Largest independent set of paired elements
elements in a solution to the matroid parity problem for this matroid are the two edges in each triangle of an optimal set of triangles. The same problem
Matroid_parity_problem
Measure of the level of acidity or basicity of an aqueous solution
to specify the acidity or basicity of aqueous solutions. Acidic solutions (solutions with higher concentrations of hydrogen (H+) cations) are measured
PH
Graph without triples of adjacent vertices
Dense triangle-free graphs are four-colorable: a solution to the Erdős–Simonovits problem (PDF). Chan, Timothy M. (2023), "Finding triangles and other
Triangle-free_graph
Astronomical technique
star x,y positions from the celestial image, groups them in three-star triangles or four-star quads. Then it calculates for each group a geometric hash
Astrometric_solving
Approximation for the travelling salesman problem
approximate solutions to the travelling salesman problem, on instances where the distances form a metric space (they are symmetric and obey the triangle inequality)
Christofides_algorithm
Construct all metric spaces where lines resemble those on a sphere
biangles, and thus be defined on triangles in the same way as the area of a triangle is defined on a sphere. Since the triangle inequality holds, it follows
Hilbert's_fourth_problem
Indian mathematician and astronomer (598–668)
essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with
Brahmagupta
Geometric treatise by Archimedes
of the blue triangle; each of the 2 3 = 8 {\displaystyle 2^{3}=8} red triangles has 1 8 {\displaystyle {\tfrac {1}{8}}} the area of a yellow triangle
Quadrature_of_the_Parabola
17th-century conjecture proved by Andrew Wiles in 1994
case n = −2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer
Fermat's_Last_Theorem
"2-comparability", or of finding compatible sets of triangles in a convex polygon. The best lower bound known for the number of tripods that can have
Tripod_packing
Area of geometry, about angles and lengths
(trígōnon) 'triangle' and μέτρον (métron) 'measure') is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In
Trigonometry
Mathematical proof technique using contradiction
r^{4}+s^{4}=t^{2}} cannot have non-trivial solutions, since non-trivial solutions would give Pythagorean triangles with two sides being squares. For other
Proof_by_infinite_descent
in each of their corners. The object of the game is, for any arrangement of the pins, to choose the positions and orientations of the triangles so that
Icosoku
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES
SOLUTION OF-TRIANGLES