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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 A are all positive integers. Heronian triangles are named
Heronian_triangle
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
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
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
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
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
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
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
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
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
once observed that the first five rows of Pascal's triangle, when read as the digits of an integer, are the corresponding powers of eleven. He claimed
Pascal's_triangle
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
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
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
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, 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
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
Heronian triangle Integer triangle Pythagorean triple Eisenstein triple Bermuda Triangle Historic Triangle Lithium Triangle Parliamentary Triangle, Canberra
List_of_triangle_topics
Integer Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A007318 (Pascal's triangle read by rows)". The On-Line Encyclopedia of Integer Sequences
1000_(number)
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
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
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
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
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
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
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
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'_theorem
Number, approximately 1.618
triangle formed by two diagonals and a side of a regular pentagon is called a golden triangle or sublime triangle. It is an acute isosceles triangle with
Golden_ratio
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
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
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
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
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
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
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)
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)
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
Integer sequence
Gould's sequence is an integer sequence named after Henry W. Gould that counts how many odd numbers are in each row of Pascal's triangle. It consists only
Gould's_sequence
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
(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)
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
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
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)
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
Property of geometry, also used to generalize the notion of "distance" in metric spaces
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
Triangle_inequality
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
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
Size of a two-dimensional surface
integer sides, integer diagonals, and integer area. Equiareal map Heronian triangle, a triangle with integer sides and integer area. List of triangle
Area
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)
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)
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
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
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)
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
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)
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
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
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
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
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 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
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
(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)
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
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
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
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)
Branch of pure mathematics
rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions
Number_theory
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)
Equation of the form 1/a + 1/b = 1/c
or not the numbers are integers; there is a formula (see here) that generates all integer cases. Second, also in a right triangle the sum of the squared
Optic_equation
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
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
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
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
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. Retrieved 2016-06-02. Sloane, N. J. A. (ed.). "Sequence A111808 (Left half of trinomial triangle (A027907)
800_(number)
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
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
Natural number
(ed.). "Sequence A000979 (Wagstaff primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-30. Sloane, N. J. A. (ed
43_(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
Algebraic expansion of powers of a binomial
nonnegative integers satisfying k + m = n {\displaystyle k+m=n} and the coefficient a {\displaystyle a} of each term is a specific positive integer depending
Binomial_theorem
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)
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
Natural number
4,294,967,295 is the highest unsigned (that is, not negative) 32-bit integer, which makes it the highest possible number a 32-bit system can store in
4,294,967,295
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
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
(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
polygonal chains of length six using horizontal and vertical segments of the integer lattice. 147 (disambiguation) Sloane, N. J. A. (ed.). "Sequence A005902
147_(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)
File format for 3D printing and scanning
little-endian unsigned integer indicating the number of triangular facets in the file. Following that is data describing each triangle in turn. The file simply
STL_(file_format)
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
Triangular array of natural numbers
k\leq n} form a triangular array of natural numbers, called the Narayana triangle, that occur in various counting problems. They are named after Canadian
Narayana_number
Natural number
Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that, when multiplied by any of the
120_(number)
On point sets with no small-area triangles
n\times n} integer grid (for 0 ≤ i < n {\displaystyle 0\leq i<n} ) has no three collinear points, and therefore by Pick's theorem each of the triangles formed
Heilbronn_triangle_problem
Invariant measure of fractal dimension
shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological
Hausdorff_dimension
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE
INTEGER TRIANGLE