Integer triangle, the Glossary

Table of Contents

1. 67 relations: Absolute value, Acute and obtuse triangles, Alcuin's sequence, Altitude (triangle), Angle, Area, Arithmetic progression, Bisection, Brahmagupta triangle, Cartesian coordinate system, Circumcircle, Congruence (geometry), Coprime integers, Eisenstein triple, Elliptic curve, Equilateral triangle, Euler brick, Euler's theorem in geometry, Extended side, Face (geometry), Geometric progression, Golden ratio, Greatest common divisor, Half-integer, Harmonic progression (mathematics), Heron's formula, Heronian triangle, History of the Theory of Numbers, Hypotenuse, Idoneal number, If and only if, Incenter, Incircle and excircles, Integer, Integer partition, Integer sequence, Isosceles triangle, Lattice (group), Lattice graph, Law of cosines, Leonard Eugene Dickson, Lowest common denominator, Median (geometry), Modular arithmetic, Niven's theorem, Parity (mathematics), Perimeter, Pick's theorem, Prime number, Pythagorean Triangles, ... Expand index (17 more) »

2. Arithmetic problems of plane geometry
3. Squares in number theory
4. Types of triangles

Absolute value

In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign.

See Integer triangle and Absolute value

Acute and obtuse triangles

An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). Integer triangle and acute and obtuse triangles are Types of triangles.

See Integer triangle and Acute and obtuse triangles

Alcuin's sequence

In mathematics, Alcuin's sequence, named after Alcuin of York, is the sequence of coefficients of the power-series expansion of: The sequence begins with these integers: The nth term is the number of triangles with integer sides and perimeter n.

See Integer triangle and Alcuin's sequence

Altitude (triangle)

In geometry, an altitude of a triangle is a line segment through a vertex and perpendicular to a line containing the side opposite the vertex.

See Integer triangle and Altitude (triangle)

Angle

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

See Integer triangle and Angle

Area

Area is the measure of a region's size on a surface.

See Integer triangle and Area

Arithmetic progression

An arithmetic progression or arithmetic sequence (AP) is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence.

See Integer triangle and Arithmetic progression

Bisection

In geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size).

See Integer triangle and Bisection

Brahmagupta triangle

A Brahmagupta triangle is a triangle whose side lengths are consecutive positive integers and area is a positive integer. Integer triangle and Brahmagupta triangle are arithmetic problems of plane geometry and Types of triangles.

See Integer triangle and Brahmagupta triangle

Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

See Integer triangle and Cartesian coordinate system

Circumcircle

In geometry, the circumscribed circle or circumcircle of a triangle is a circle that passes through all three vertices.

See Integer triangle and Circumcircle

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.

See Integer triangle and Congruence (geometry)

Coprime integers

In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1.

See Integer triangle and Coprime integers

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. Integer triangle and Eisenstein triple are arithmetic problems of plane geometry.

See Integer triangle and Eisenstein triple

Elliptic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point.

See Integer triangle and Elliptic curve

Equilateral triangle

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. Integer triangle and equilateral triangle are Types of triangles.

See Integer triangle and Equilateral triangle

Euler brick

In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths.

See Integer triangle and Euler brick

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.

See Integer triangle and Euler's theorem in geometry

Extended side

In plane geometry, an extended side or sideline of a polygon is the line that contains one side of the polygon.

See Integer triangle and Extended side

Face (geometry)

In solid geometry, a face is a flat surface (a planar region) that forms part of the boundary of a solid object; a three-dimensional solid bounded exclusively by faces is a polyhedron.

See Integer triangle and Face (geometry)

Geometric progression

A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

See Integer triangle and Geometric progression

Golden ratio

In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

See Integer triangle and Golden ratio

Greatest common divisor

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

See Integer triangle and Greatest common divisor

Half-integer

In mathematics, a half-integer is a number of the form n + \tfrac, where n is an integer.

See Integer triangle and Half-integer

Harmonic progression (mathematics)

In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression.

See Integer triangle and Harmonic progression (mathematics)

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 Letting be the semiperimeter of the triangle, s.

See Integer triangle and Heron's formula

Heronian triangle

In geometry, a Heronian triangle (or Heron triangle) is a triangle whose side lengths,, and and area are all positive integers. Integer triangle and Heronian triangle are arithmetic problems of plane geometry and Types of triangles.

See Integer triangle and Heronian triangle

History of the Theory of Numbers

History of the Theory of Numbers is a three-volume work by Leonard Eugene Dickson summarizing work in number theory up to about 1920. Integer triangle and History of the Theory of Numbers are squares in number theory.

See Integer triangle and History of the Theory of Numbers

Hypotenuse

In geometry, a hypotenuse is the side of a right triangle opposite the right angle.

See Integer triangle and Hypotenuse

Idoneal number

In mathematics, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers D such that any integer expressible in only one way as x2 ± Dy2 (where x2 is relatively prime to Dy2) is a prime power or twice a prime power.

See Integer triangle and Idoneal number

If and only if

In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements.

See Integer triangle and If and only if

Incenter

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale.

See Integer triangle and Incenter

Incircle and excircles

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.

See Integer triangle and Incircle and excircles

Integer

An integer is the number zero (0), a positive natural number (1, 2, 3,...), or the negation of a positive natural number (−1, −2, −3,...). The negations or additive inverses of the positive natural numbers are referred to as negative integers.

See Integer triangle and Integer

Integer partition

In number theory and combinatorics, a partition of a non-negative integer, also called an integer partition, is a way of writing as a sum of positive integers.

See Integer triangle and Integer partition

Integer sequence

In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.

See Integer triangle and Integer sequence

Isosceles triangle

In geometry, an isosceles triangle is a triangle that has two sides of equal length. Integer triangle and isosceles triangle are Types of triangles.

See Integer triangle and Isosceles triangle

Lattice (group)

In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point.

See Integer triangle and Lattice (group)

Lattice graph

In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space, forms a regular tiling.

See Integer triangle and Lattice graph

Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.

See Integer triangle and Law of cosines

Leonard Eugene Dickson

Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician.

See Integer triangle and Leonard Eugene Dickson

Lowest common denominator

In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions.

See Integer triangle and Lowest common denominator

In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side.

See Integer triangle and Median (geometry)

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus.

See Integer triangle and Modular arithmetic

Niven's theorem

In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: \begin \sin 0^\circ &.

See Integer triangle and Niven's theorem

Parity (mathematics)

In mathematics, parity is the property of an integer of whether it is even or odd.

See Integer triangle and Parity (mathematics)

Perimeter

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length.

See Integer triangle and Perimeter

Pick's theorem

In geometry, Pick's theorem provides a formula for the area of a simple polygon with integer vertex coordinates, in terms of the number of integer points within it and on its boundary.

See Integer triangle and Pick's theorem

Prime number

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

See Integer triangle and Prime number

Pythagorean Triangles

Pythagorean Triangles is a book on right triangles, the Pythagorean theorem, and Pythagorean triples.

See Integer triangle and Pythagorean Triangles

Pythagorean triple

A Pythagorean triple consists of three positive integers,, and, such that. Integer triangle and Pythagorean triple are arithmetic problems of plane geometry and squares in number theory.

See Integer triangle and Pythagorean triple

Rational number

In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

See Integer triangle and Rational number

Right triangle

A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular forming a right angle (turn or 90 degrees). Integer triangle and right triangle are Types of triangles.

See Integer triangle and Right triangle

Robbins pentagon

In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers. Integer triangle and Robbins pentagon are arithmetic problems of plane geometry.

See Integer triangle and Robbins pentagon

Scaling (geometry)

In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a scale factor that is the same in all directions.

See Integer triangle and Scaling (geometry)

Semiperimeter

In geometry, the semiperimeter of a polygon is half its perimeter.

See Integer triangle and Semiperimeter

Similarity (geometry)

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other.

See Integer triangle and Similarity (geometry)

Sine and cosine

In mathematics, sine and cosine are trigonometric functions of an angle.

See Integer triangle and Sine and cosine

Square number

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. Integer triangle and square number are squares in number theory.

See Integer triangle and Square number

Square root

In mathematics, a square root of a number is a number such that y^2.

See Integer triangle and Square root

Tetrahedron

In geometry, a tetrahedron (tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices.

See Integer triangle and Tetrahedron

The Mathematical Gazette

The Mathematical Gazette is a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association.

See Integer triangle and The Mathematical Gazette

Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry.

See Integer triangle and 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.

See Integer triangle and Triangle inequality

Triangular number

A triangular number or triangle number counts objects arranged in an equilateral triangle. Integer triangle and triangular number are squares in number theory.

See Integer triangle and Triangular number

Up to

Two mathematical objects and are called "equal up to an equivalence relation ".

See Integer triangle and Up to

Volume

Volume is a measure of regions in three-dimensional space.

See Integer triangle and Volume

See also

Arithmetic problems of plane geometry

• Brahmagupta triangle
• Congruent number
• Eisenstein triple
• Erdős–Anning theorem
• Erdős–Diophantine graph
• Erdős–Ulam problem
• Fermat's right triangle theorem
• Harborth's conjecture
• Heronian triangle
• Integer triangle
• Pythagorean quadruple
• Pythagorean triple
• Robbins pentagon

Squares in number theory

• Büchi's problem
• Basel problem
• Brahmagupta–Fibonacci identity
• Brocard's conjecture
• Congruence of squares
• Congruum
• Degen's eight-square identity
• Dixon's factorization method
• Euler's criterion
• Euler's four-square identity
• Fermat's right triangle theorem
• Fermat's theorem on sums of two squares
• Gauss's lemma (number theory)
• History of the Theory of Numbers
• Integer triangle
• Jacobi's four-square theorem
• Lagrange's four-square theorem
• Legendre's conjecture
• Legendre's three-square theorem
• Olga Taussky-Todd
• Pythagorean prime
• Pythagorean quadruple
• Pythagorean triple
• Quadratic form
• Ramanujan's sum
• Ramanujan's ternary quadratic form
• Square (algebra)
• Square number
• Sum of squares function
• Sum of two squares theorem
• Sums of powers
• Sun Zhiwei
• The Book of Squares
• Triangular number
• Unit square
• Waring's problem
• Zolotarev's lemma

Types of triangles

• Acute and obtuse triangles
• Automedian triangle
• Brahmagupta triangle
• Calabi triangle
• Circular triangle
• Equilateral triangle
• Golden triangle (mathematics)
• Heptagonal triangle
• Heronian triangle
• Hyperbolic triangle
• Ideal triangle
• Integer triangle
• Isosceles triangle
• Kepler triangle
• Reuleaux triangle
• Right triangle
• Sierpiński triangle
• Skinny triangle
• Special right triangle

References

[1] https://en.wikipedia.org/wiki/Integer_triangle

Also known as Integer geometry, Integer sided triangle, Integer-sided triangle, Integral triangle, Rational triangle, Rational triangles.

, Pythagorean triple, Rational number, Right triangle, Robbins pentagon, Scaling (geometry), Semiperimeter, Similarity (geometry), Sine and cosine, Square number, Square root, Tetrahedron, The Mathematical Gazette, Triangle, Triangle inequality, Triangular number, Up to, Volume.