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Triangle with integer side lengths
An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational
Integer_triangle
Integer side lengths of a right triangle
for any positive integer k. A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean
Pythagorean_triple
Triangle whose side lengths and area are integers
Heronian triangle (or Heron triangle) is a triangle whose side lengths a, b, and c and area T are all positive integers. Heronian triangles are named
Heronian_triangle
Right triangle with a feature making calculations on the triangle easier
ratio of two integers. However, infinitely many almost-isosceles right triangles do exist. These are right-angled triangles with integer sides for which
Special_right_triangle
Figurate number
numbers or triangle numbers are the sequence of positive integers that can be represented as a lattice of points arranged in an equilateral triangle. The triangular
Triangular_number
Array of partial sums of the binomial coefficients
Bernoulli's triangle is an array of partial sums of the binomial coefficients. For any non-negative integer n and for any integer k included between 0
Bernoulli's_triangle
Division of something into two equal or congruent parts
} No two non-congruent triangles share the same set of three internal angle bisector lengths. There exist integer triangles with a rational angle bisector
Bisection
Formula for area of a grid polygon
the polygon into triangles with three integer vertices and no other integer points. One can then prove that each subdivided triangle has area exactly
Pick's_theorem
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Heronian triangle Integer triangle Pythagorean triple Eisenstein triple Bermuda Triangle Historic Triangle Lithium Triangle Parliamentary Triangle, Canberra
List_of_triangle_topics
Set of integers, the lengths of the sides of a triangle with a 60° angle
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 Eisenstein
Eisenstein_triple
Polygon mesh composed of triangles
are[citation needed] Vertex = <integer>; // v Edge = <integer, integer>; // v0, v1 Triangle <integer,integer,integer>; // v0, v1, v2 VData = <application-specific
Triangle_mesh
Triangles without a right angle
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
Acute_and_obtuse_triangles
Triangle containing a 90-degree angle
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular
Right_triangle
Triangular array of the binomial coefficients
Pascal's triangle: (20)=1, (21)=2, (22)=1. In general, the binomial theorem states that when a binomial like x+y is raised to a positive integer power n
Pascal's_triangle
Natural number
(Hypotenuses for which there exist exactly 3 distinct integer triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed
1000_(number)
Curved triangle with constant width
A Reuleaux triangle [ʁœlo] is a curved triangle with constant width, the simplest and best known curve of constant width other than the circle. It is formed
Reuleaux_triangle
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Fractal composed of triangles
can be interpreted as a discrete analogue of the Sierpiński triangle. For objects of integer dimension d {\displaystyle d} , scaling a figure by a factor
Sierpiński_triangle
Triangle with specific characteristics
A Brahmagupta triangle is a triangle whose side lengths are consecutive positive integers and area is a positive integer. The triangle whose side lengths
Brahmagupta_triangle
Rational right triangles cannot have square area
do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle. More abstractly
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
There is a primitive integer-sided automedian triangle that is not generated from a right triangle: namely, the equilateral triangle with sides of unit
Automedian_triangle
Shape with three equal sides
triangle is a triangle with three sides of equal length and three equal angles. It is a regular polygon, occasionally known as the regular triangle.
Equilateral_triangle
Special ordering of numbers on a triangle
A magic triangle is a magic arrangement of the integers from 1 to n in a triangular figure. A magic triangle or perimeter magic triangle is an arrangement
Magic_triangle_(mathematics)
Topics referred to by the same term
group Primitive root of unity; See Root of unity Primitive triangle, an integer triangle whose sides have no common prime factor Primitive (phylogenetics)
Primitive
Triangle of numbers
In a format similar to that of Pascal's triangle, and in the order listed in the On-Line Encyclopedia of Integer Sequences (OEIS), its first few rows are:
Bell_triangle
Relation between sides of a right triangle
root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem
Pythagorean_theorem
Overview of and topical guide to geometry
trapezoid Triangle Acute and obtuse triangles Equilateral triangle Euler's line Heron's formula Integer triangle Heronian triangle Isosceles triangle List
Outline_of_geometry
Number, approximately 2.41421
^{2}&=-1+2\sum _{n=0}^{\infty }(\sigma -1)^{-n}.\end{aligned}}} For every integer n {\displaystyle n} one has σ n = 2 σ n − 1 + σ n − 2 = σ n − 1 + 3
Silver_ratio
Natural number
)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001263 (Triangle of Narayana numbers T(n,k) = C(n-1
50_(number)
Number, approximately 1.618
φ. The ring of integers in this field, ℤ[φ], sometimes called the golden integers, are numbers of the form a+bφ with integer coefficients a and
Golden_ratio
Triangle with at least two sides congruent
For any integer n ≥ 4 {\displaystyle n\geq 4} , any triangle can be triangulated into n {\displaystyle n} isosceles triangles. In a right triangle, the median
Isosceles_triangle
Number of subsets of a given size
the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and
Binomial_coefficient
Cuboid whose edges and face diagonals have integer lengths
Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are
Euler_brick
Equation for radii of tangent circles
quadruples of this type are also closely related to Heronian triangles, triangles with integer sides and area. Starting with any four mutually tangent circles
Descartes's_theorem
Property of geometry, also used to generalize the notion of "distance" in metric spaces
In mathematics, the triangle inequality states that for any triangle in Euclidean space, the sum of the lengths of any two sides must be greater than or
Triangle_inequality
Perpendicular line segment from a triangle's side to opposite vertex
formula for the reciprocal area of a triangle", Mathematical Gazette 89 (November 2005), 494. Voles, Roger, "Integer solutions of a − 2 + b − 2 = d − 2
Altitude_(triangle)
Points on a common circle
and all of their side lengths are positive integers. Triangles with this property are called Heronian triangles; cyclic quadrilaterals with this property
Concyclic_points
Sloane, N. J. A. (ed.), "Sequence A008284 (Triangle of partition numbers)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation Arndt, Jörg (2011)
Triangle_of_partition_numbers
On sets of points with integer distances
the even multiples of one of the acute angles of an integer-sided right triangle (such as the triangle with side lengths 3, 4, and 5) has this property.
Erdős–Anning_theorem
Natural number
the OEIS). It forms one side in a primitive Pythagorean triangle (right triangles with integer sides). It is also a number "k" such that 4*k+1 is the hypotenuse
51_(number)
Conjecture in combinatorial number theory
Erdős, and Hanson (1974), the number of integers no larger than x that appear more than twice in Pascal's triangle is O ( x 1 / 2 ) {\displaystyle O(x^{1/2})}
Singmaster's_conjecture
Shape with the same area and perimeter
Pythagorean triples describing integer-sided right triangles, and there are infinitely many equable right triangles with non-integer sides; however, there are
Equable_shape
Right triangle related to the golden ratio
be adequately explained using integer ratios, based on a right triangle with sides 11 and 14. The name "Kepler triangle" for this shape was used by Roger
Kepler_triangle
Graph whose embedding in a Euclidean space forms a regular tiling
the wazir form a square lattice graph. Lattice path Pick's theorem Integer triangles in a 2D lattice Regular graph Weisstein, Eric W. "Lattice graph".
Lattice_graph
Natural number
number preceding a cube. As an early prime number in the series of positive integers, the number seven has symbolic associations in religion, mythology, superstition
7
Natural number
only prime triangular number, and Carl Friedrich Gauss proved that every integer is the sum of at most three triangular numbers. Three is the only prime
3
Numbers arranged in a triangle
triangle, which counts permutations with a given number of ascents Floyd's triangle, whose entries are all of the integers in order Hosoya's triangle
Triangular_array
Natural number
(Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2))". The On-Line Encyclopedia of Integer Sequences
400_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2021-09-24. Sloane, N. J. A. (ed.). "Sequence A006785 (Number of triangle-free graphs on
107_(number)
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A006785 (Number of triangle-free graphs on n vertices)"
1,000,000
Relation between the side lengths and altitude of a right triangle
is as follows: Let A, B be the endpoints of the hypotenuse of a right triangle △ABC. Let D be the foot of a perpendicular dropped from C, the vertex of
Inverse_Pythagorean_theorem
Natural number
(Smallest side lengths of almost-equilateral Heronian triangles)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 22 May 2022.
700_(number)
On graph drawing with integer edge lengths
rational distances must grow arbitrarily large. Integer triangle, an integral Fáry embedding of the triangle graph Matchstick graph, a graph that can be drawn
Harborth's_conjecture
Number whose square is a given number
roots of positive integers that are not perfect squares are always irrational numbers: numbers not expressible as a ratio of two integers (that is, they
Square_root
Size of a two-dimensional surface
Heronian triangle, a triangle with integer sides and integer area. List of triangle inequalities One-seventh area triangle, an inner triangle with one-seventh
Area
Mathematics term
{k-1}{2}}\right)} of Pascal's triangle. The diagonals next to the edge diagonals contain the positive integers in order, but with each integer stated twice (OEIS: A004526)
Lozanić's_triangle
17th-century conjecture proved by Andrew Wiles in 1994
coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangle and d is the integer altitude
Fermat's_Last_Theorem
Mathematical proof technique using contradiction
right triangle with legs y z {\displaystyle {\sqrt {yz}}} and b 2 {\displaystyle b^{2}} and hypotenuse a 2 {\displaystyle a^{2}} also would have integer sides
Proof_by_infinite_descent
Prime number congruent to 1 mod 4
{\displaystyle {\sqrt {p}}} is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers p {\displaystyle p} for
Pythagorean_prime
Computer markup language
000000000140000000000000004010000000000000, where: 1-byte integer 00 or 0: big endian 4-byte integer 00000001 or 1: POINT (2D) 8-byte float 4000000000000000
Well-known text representation of geometry
Well-known_text_representation_of_geometry
divisors of the numbers 1 to 1000. A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n).
Table_of_divisors
Natural number
(151 + 157). It is the totient sum of the first 41 integers. 309 = 3 × 103. It is a Blum integer and a centered icosahedral number. 310 = 2 × 5 × 31
300_(number)
Natural number
divisors (including itself) is also an integer, 54 is an arithmetic number. If the complementary angle of a triangle's corner is 54 degrees, the sine of that
54_(number)
Polynomial sequence
A. (ed.). "Sequence A060187 (Triangle read by rows: Eulerian numbers of type B)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Schoenberg
Eulerian_number
Natural number between 89 and 91
Encyclopedia of Integer Sequences. OEIS Foundation. "Sloane's A008277 :Triangle of Stirling numbers of the second kind". The On-Line Encyclopedia of Integer Sequences
90_(number)
Natural number
number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5). 5
5
Similar triangles that share two side lengths
exercises about triangles. There are infinitely many pairs of 5-Con triangles, even up to scaling. The smallest 5-Con triangles with integer sides have side
5-Con_triangles
Natural number
summatory function Φ ( n ) {\displaystyle \Phi (n)} over the first 10 integers, and the smallest number n {\displaystyle n} with exactly 7 solutions for
32_(number)
Tetrahedron where all three face angles at one vertex are right angles
called the altitude of the tetrahedron (analogous to the altitude of a triangle). An example of a trirectangular tetrahedron is a truncated solid figure
Trirectangular_tetrahedron
Point in a triangle that can be seen as its middle under some criteria
{\displaystyle f(a,b,c)=1} otherwise. Then for any triangle with integer sides the associated triangle center evaluates to 0:0:0 which is undefined. In
Triangle_center
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A094133 (Leyland prime numbers)". The On-Line Encyclopedia of Integer Sequences
17_(number)
Natural number
A138591 (Sums of two or more consecutive nonnegative integers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-04-20. Sloane
777_(number)
Algebraic construction
of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality
Ring_of_integers
Natural number
arithmetic number. 69 is a congruent number—a positive integer that is the area of a right triangle with three rational number sides—and an amenable number
69_(number)
Functions of an angle
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 sciences that
Trigonometric_functions
Number system extending the rational numbers
integer (possibly negative), and each a i {\displaystyle a_{i}} is an integer such that 0 ≤ a i < p . {\displaystyle 0\leq a_{i}<p.} A p-adic integer
P-adic_number
Natural number
having 3+3n vertices of a triangle with each side subdivided by n additional points)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
229_(number)
Natural number
unlabeled nodes. If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices
112_(number)
Natural number
about the integer". mathworld.wolfram.com. "Refactorable numbers". mathworld.wolfram.com. Sloane, N. J. A. (ed.). "Sequence A331452 (Triangle read by rows:
296_(number)
Natural number
sequence". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. "Sloane's A008277 : Triangle of Stirling numbers of the second
65_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-02. Sloane, N. J. A. (ed.). "Sequence A111808 (Left half of trinomial triangle (A027907)
800_(number)
Natural number
eighth triangular number or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number. Aside from
36_(number)
Natural number
(Conjecturally largest even integer which is an unordered sum of two primes in exactly n ways)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation
68_(number)
Natural number
triangular numbers together. This sequence, dubbed 'Triple Triangle-Pair Numbers' is the sequence of integers: 276, 406, 666, ... 276 is the size of the largest
276_(number)
Natural number
Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences
45_(number)
Natural number
(ed.). "Sequence A006785 (Number of triangle-free graphs on n vertices)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N
100,000
Natural number
numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A331452 (Triangle read by rows: T(n,m) (n >= m >=
600_(number)
Natural number
primitive Heronian tetrahedron with 4 congruent triangle faces.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2024-03-09
888_(number)
holds whether or not the numbers are integers; there is a formula (see here) that generates all integer cases. A triangle not necessarily in the Euclidean
List_of_sums_of_reciprocals
Mathematical transformation on sequences
will be exactly N {\displaystyle N} rows in the triangle, such that k {\displaystyle k} is an integer in the range: [ 0 , N ) {\displaystyle [0,N)} (exclusive)
Boustrophedon_transform
Number of paths between grid corners, allowing diagonal steps
{\displaystyle m} and n {\displaystyle n} , the points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and
Delannoy_number
Circle with radius of one
+ θ ) {\displaystyle \sin \theta =\sin(2\pi k+\theta )} for any integer k. Triangles constructed on the unit circle can also be used to illustrate the
Unit_circle
Shape with four equal sides and angles
area is the square root of the area. Squaring an integer, or taking the area of a square with integer sides, results in a square number; these are figurate
Square
Natural number
235 (two hundred [and] thirty-five) is the integer following 234 and preceding 236. 235 is a heptagonal number, a centered triangular number, and a Smarandache–Wellin
235_(number)
Natural number
The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A006785 (Number of triangle-free graphs on n vertices)"
100,000,000
Polynomial equation whose integer solutions are sought
a Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation
Diophantine_equation
Index of articles associated with the same name
Pythagorean primes. Pythagorean triangles with integer altitude from the hypotenuse have the sum of squares of inverses of the integer legs equal to the square
Sum_of_squares
Type of complex number
a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio
Algebraic_number
Fundamental trigonometric functions
acute angle are defined in the context of a right triangle: the sine of an acute angle of a right triangle is the ratio of the length of the side opposite
Sine_and_cosine
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INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
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