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Integer side lengths of a right triangle
a^{2}+b^{2}=c^{2}} ; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides
Pythagorean_triple
Relation between sides of a right triangle
objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. Forms of the Pythagorean theorem have appeared
Pythagorean_theorem
Book about right triangles by Wacław Sierpiński in 1954
Pythagorean Triangles is a book on right triangles, the Pythagorean theorem, and Pythagorean triples. It was originally written in the Polish language
Pythagorean_Triangles
Triangle containing a 90-degree angle
a+b+c=r+r_{a}+r_{b}+r_{c}.} Acute and obtuse triangles (oblique triangles) Spiral of Theodorus Trirectangular spherical triangle Artmann, Benno (2012) [1999], Euclid:
Right_triangle
Right triangle with a feature making calculations on the triangle easier
several Pythagorean triples which are well-known, including those with sides in the ratios: The 3 : 4 : 5 triangles are the only right triangles with edges
Special_right_triangle
Triangle with integer side lengths
relationship between integer triangles and rational triangles. Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and
Integer_triangle
Prime number congruent to 1 mod 4
primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; 5 {\displaystyle {\sqrt {5}}} is the hypotenuse of a right triangle with
Pythagorean_prime
Four integers where the sum of the squares of three equals the square of the fourth
of areas of this class of Heronian triangles can be found at (sequence A367737 in the OEIS). A primitive Pythagorean quadruple (a, b, c, d) parametrized
Pythagorean_quadruple
Mathematical proof by James Garfield
Garfield's proof of the Pythagorean theorem is an original proof of the Pythagorean theorem discovered by James A. Garfield, the 20th president of the
Garfield's proof of the Pythagorean theorem
Garfield's_proof_of_the_Pythagorean_theorem
Triangle whose side lengths and area are integers
area of each Pythagorean triangle is an integer). There are Heronian triangles that cannot be obtained by joining Pythagorean triangles. For example,
Heronian_triangle
Unsolved problem about sums of powers
b2 + c2 = d2 + e2 + f2 Thus, this equation seems to contain two Pythagorean Triangles. from equation a1 + b1 + c1 = d1 + e1 + f1 => a1 + b1 - d1 = e1
Prouhet–Tarry–Escott_problem
Besides Euclid's formula, many other formulas for generating Pythagorean triples have been developed. Euclid's, Pythagoras' and Plato's formulas for calculating
Formulas for generating Pythagorean triples
Formulas_for_generating_Pythagorean_triples
Fractal composed of triangles
Sierpinski Triangle in 3D Pythagorean triangles, Waclaw Sierpinski, Courier Corporation, 2003 A067771 Number of vertices in Sierpiński triangle of order
Sierpiński_triangle
Rational right triangles cannot have square area
right triangles that share two sides in this way. Because the congrua are exactly the numbers that are four times the area of a Pythagorean triangle, and
Fermat's right triangle theorem
Fermat's_right_triangle_theorem
Philosophical system based on the teachings of Pythagoras
Pythagoreanism originated in the 6th century BC, based on and around the teachings and beliefs held by Pythagoras and his followers, the Pythagoreans
Pythagoreanism
Relation between sine and cosine
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric
Pythagorean trigonometric identity
Pythagorean_trigonometric_identity
Longest side of a right-angled triangle, the side opposite of the right angle
{a^{2}+b^{2}}}} . This is sometimes known as Pythagorean addition. For example, if the two legs of a right triangle have lengths 3 and 4, respectively, then
Hypotenuse
Generalization of Pythagorean theorem
\end{aligned}}} The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if γ {\displaystyle \gamma } is a right angle
Law_of_cosines
Generalization of golden and silver ratios
\theta } is a positive integer, as it is with some Pythagorean triangles. For a primitive Pythagorean triple, a2 + b2 = c2, with positive integers a < b
Metallic_mean
Relation between the side lengths and altitude of a right triangle
about right triangles Pythagorean theorem – Relation between sides of a right triangle R. B. Nelsen, Proof Without Words: A Reciprocal Pythagorean Theorem
Inverse_Pythagorean_theorem
Perpendicular line segment from a triangle's side to opposite vertex
inverse Pythagorean theorem) For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with
Altitude_(triangle)
Triangles without a right angle
Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are
Acute_and_obtuse_triangles
Index of articles associated with the same name
theorem on sums of two squares states which primes are Pythagorean primes. Pythagorean triangles with integer altitude from the hypotenuse have the sum
Sum_of_squares
Length of a line segment
Cartesian coordinates of the points using the Pythagorean theorem, and therefore is occasionally called the Pythagorean distance. These names come from the ancient
Euclidean_distance
Mathematical proof technique using contradiction
than 1) Pythagorean triangle with the same property. Primitive Pythagorean triangles' sides can be written as x = 2 a b , {\displaystyle x=2ab,} y =
Proof_by_infinite_descent
Natural number
ways, 85 = 92 + 22 = 72 + 62. the length of the hypotenuse of four Pythagorean triangles. a Smith number in decimal. The radix of the Ascii85 (sometimes
85_(number)
Right triangle related to the golden ratio
same triangle characterize it in terms of the three Pythagorean means of two numbers, or via the inradius of isosceles triangles. This triangle is named
Kepler_triangle
Set of integers, the lengths of the sides of a triangle with a 60° angle
relation of such triangles to the Eisenstein integers is analogous to the relation of Pythagorean triples to the Gaussian integers. Triangles with an angle
Eisenstein_triple
diagram Triangle mesh Nonobtuse mesh Encyclopedia of Triangle Centers Pythagorean Triangles The Secrets of Triangles Triangular matrix (2,3,7) triangle group
List_of_triangle_topics
Numbers obtained by adding the two previous ones
(F_{n}F_{n+3})^{2}+(2F_{n+1}F_{n+2})^{2}={F_{2n+3}}^{2}.} The sequence of Pythagorean triangles obtained from this formula has sides of lengths (3,4,5), (5,12,13)
Fibonacci_sequence
Spacing between equally-spaced square numbers
this formula, each congruum is four times the area of a Pythagorean triangle, a right triangle whose sides are integers. Congrua are also closely connected
Congruum
Germain prime, super-prime 1500 = hypotenuse in three different Pythagorean triangles 1501 = centered pentagonal number 1502 = number of pairs of consecutive
1000_(number)
Polyhedron with four faces
tetrahedron. A disphenoid is a tetrahedron with four congruent triangles as faces; the triangles necessarily have all angles acute. The regular tetrahedron
Tetrahedron
Polish mathematician (1882–1969)
English by Canadian mathematician Cecilia Krieger. Another book, Pythagorean Triangles (1954), was translated into English by Indian mathematician Ambikeshwar
Wacław_Sierpiński
a permutation thereof, analogous to the Pythagorean theorem characterizing right triangles as the triangles satisfying the formula a 2 + b 2 = c 2 {\displaystyle
Automedian_triangle
Unique positive real number which when multiplied by itself gives 2
1090/conm/039/788163. ISBN 0821850407. ISSN 0271-4132. Sierpiński, Wacław (2003). Pythagorean Triangles. Translated by Sharma, Ambikeshwa. Mineola, NY: Dover. pp. 4–6.
Square_root_of_2
Relates the tangent of half of an angle to trigonometric functions of the entire angle
third angle is a right angle then a triangle with these interior angles can be scaled to a Pythagorean triangle. If the third angle is not required to
Tangent_half-angle_formula
Babylonian clay tablet of numbers in Pythagorean triples
triangles, are possible. The purpose of Plimpton 322 is not known. Neugebauer and Sachs saw Plimpton 322 as a study of solutions to the Pythagorean equation
Plimpton_322
Natural number
33 + 42 + 51. 65 is the length of the hypotenuse of 4 different Pythagorean triangles, the lowest number to have more than 2: 652 = 162 + 632 = 332 +
65_(number)
Property of geometry, also used to generalize the notion of "distance" in metric spaces
triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may
Triangle_inequality
Conjecture in number theory
27 (6): 513–43. doi:10.1112/blms/27.6.513. Wacław Sierpiński, Pythagorean Triangles, Dover, 2003, p. 55 (orig. Graduate School of Science, Yeshiva University
Beal_conjecture
Heronian triangle Pythagorean triangle Isosceles heronian triangle Primitive Heronian triangle Right triangle 30-60-90 triangle Isosceles right triangle Kepler
List of two-dimensional geometric shapes
List_of_two-dimensional_geometric_shapes
Type of spline curve
In mathematics, a Pythagorean hodograph curve or PH curve is a curve defined by a polynomial parametric equation for which the speed (the derivative of
Pythagorean_hodograph_curve
Overview of and topical guide to geometry
of triangle inequalities List of triangle topics Pedal triangle Pedoe's inequality Pythagorean theorem Pythagorean triangle Right triangle Triangle inequality
Outline_of_geometry
Greek philosopher (c. 570 – c. 495 BC)
ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia
Pythagoras
Mathematical tree of integer right triangles
primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean triple is
Tree of primitive Pythagorean triples
Tree_of_primitive_Pythagorean_triples
Iron Age culture of Europe
centre of Bibracte has a sophisticated geometric design based on Pythagorean triangles and incorporates an astronomical alignment, indicating that it may
La_Tène_culture
Polygonal curve made from right triangles
(also called the square root spiral, Pythagorean spiral, or Pythagoras's snail) is a spiral composed of right triangles, placed edge-to-edge. It was named
Spiral_of_Theodorus
Classical averages studied in ancient Greece
In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM). These means were
Pythagorean_means
Philosophical concept of a most basic substance, or supreme being
to a most basic or original substance. As originally conceived by the Pythagoreans, the Monad is therefore Supreme Being, divinity, or the totality of all
Monad_(philosophy)
Triangle with at least two sides congruent
the Pythagorean theorem using the fact that the altitude bisects the base and partitions the isosceles triangle into two congruent right triangles. The
Isosceles_triangle
Archaeological culture in Europe
in Denmark (0.7855 m) dating from the Bronze Age (c. 1350 BC). Pythagorean triangles were likely used in building construction to create right angles
Hallstatt_culture
Symbol of ten points laid in four rows
by a triangle of three points) the fourth row represented three dimensions (a tetrahedron defined by four points) A prayer of the Pythagoreans shows
Tetractys
considered that setting out also involved the use of equilateral or Pythagorean triangles, pentagons, and octagons. Two authors believe the Golden Section
List of works designed with the golden ratio
List_of_works_designed_with_the_golden_ratio
the analog in the plane of Lexell's theorem about spherical triangles: all of the triangles with a fixed base side, a fixed area, and the apex on the same
Area_of_a_triangle
Tiling by squares of two sizes
or equilateral triangles, and three are formed from equilateral triangles and regular hexagons. The remaining one is the Pythagorean tiling. This tiling
Pythagorean_tiling
Idea for signaling extraterrestrial beings from Earth
Gauss's Pythagorean right triangle proposal is an idea attributed to Carl Friedrich Gauss for a method to signal extraterrestrial beings by constructing
Gauss's Pythagorean right triangle proposal
Gauss's_Pythagorean_right_triangle_proposal
European archaeological culture, 2800–1800 BC
both the solar and lunar cycles "in an ingenious design based on Pythagorean triangles", reflecting the geometric relations of astronomical events (such
Bell_Beaker_culture
Cuboid whose edges and face diagonals have integer lengths
congruent number elliptic curve of rank at least 2. Pythagorean quadruple Wacław Sierpiński, Pythagorean Triangles, Dover Publications, 2003 (orig. ed. 1962).
Euler_brick
Mathematical model of the physical space
propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and
Euclidean_geometry
Educational methodology
cue-cards form during the progression of the game, a right-angled or Pythagorean triangle. It is a theoretical educational method that is made up of several
Pythagorean Method of Memorization
Pythagorean_Method_of_Memorization
Illustration of the Pythagorean theorem
sense of the Greek word came to be applied to right triangles with three squares, and to the Pythagorean theorem. Arabic speakers writing in Greek would often
Bride's_Chair
Theorem about right triangles
the geometric mean theorem there are three right triangles △ABC, △ADC and △DBC in which the Pythagorean theorem yields: h 2 = a 2 − q 2 h 2 = b 2 − p 2
Geometric_mean_theorem
Hypotenuse of right triangle from its sides
mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two
Pythagorean_addition
Complex numbers with unit norm and both real and imaginary parts rational numbers
points turns out to be closely related to primitive Pythagorean triples. Consider a primitive right triangle, that is, with integer side lengths a, b, c, with
Group of rational points on the unit circle
Group_of_rational_points_on_the_unit_circle
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
Legend about the discovery of musical tuning
perhaps leads to, the Pythagorean conception of mathematics as nature's modus operandi. As Aristotle was later to write, "the Pythagoreans construct the whole
Pythagorean_hammers
Five-pointed star polygon
triangles: five acute and five obtuse isosceles triangles. In all of them, the ratio of the longer side to the shorter side is φ. The acute triangles
Pentagram
Period of British history from c. 2500 until c. 800 BC
both the solar and lunar cycles "in an ingenious design based on Pythagorean triangles", reflecting the geometric relations of astronomical events (such
Bronze_Age_Britain
Relationship between two figures of the same shape and size, or mirroring each other
side can be calculated using the Pythagorean theorem thus allowing the SSS postulate to be applied. If two triangles satisfy the SSA condition and the
Congruence_(geometry)
Circle with radius of one
|y| are the lengths of the legs of a right triangle whose hypotenuse has length 1. Thus, by the Pythagorean theorem, x and y satisfy the equation x 2 +
Unit_circle
Group of Vedic Sanskrit texts
'rectangular triangles' instead of 'oblongs'. The length of the diagonals of these oblongs or of the hypotenuses of these rectangular triangles is not explicitly
Baudhayana_sutras
Branch of mathematics
Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas). In the Middle Ages, mathematics
Geometry
Area of geometry, about angles and lengths
similar triangles and discovered some properties of these ratios but did not turn that into a systematic method for finding sides and angles of triangles. The
Trigonometry
Rectangle with side lengths in the golden ratio
adjoining right triangles, tracing a whirl of converging golden rectangles. The logarithmic spiral through the vertices of adjacent triangles has polar slope
Golden_rectangle
Functions of an angle
dividing the triangle into two right ones and using the Pythagorean theorem. The law of cosines can be used to determine a side of a triangle if two sides
Trigonometric_functions
Fundamental result in geometry
Equidistance postulate, Playfair axiom, Proclus axiom, the Triangle postulate and the Pythagorean theorem. Keith J. Devlin (2000). The Language of Mathematics:
Sum_of_angles_of_a_triangle
Field in which every sum of two squares is a square
a Pythagorean field is a field in which every sum of two squares is a square: equivalently it has a Pythagoras number equal to 1. A Pythagorean extension
Pythagorean_field
Equality of triangles between three squares
of a triangle to the area of each of the triangles formed by squares drawn along its sides. Let △ A B C {\displaystyle \triangle ABC} be a triangle in the
Cross's_theorem
Madrasa in Aleppo, Syria
3-4-5 ratio in which a rectangular area was made by combining two pythagorean triangles. The heights of the building's columns shared the same measurement
Al-Firdaws_Madrasa
Property of objects which are scaled or mirrored versions of each other
theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles also provide the foundations for right triangle trigonometry. The concept of similarity
Similarity_(geometry)
Gallic fortified town and capital of the Aedui
circles intersecting at 1/5 of their diameter, forming a precise 3:4:5 Pythagorean triangle between the centre of the circles, the centre of the basin and the
Bibracte
Triangle area in terms of side lengths
incenter and one excircle of the triangle, or as a special case of De Gua's theorem (for the particular case of acute triangles), or as a special case of Brahmagupta's
Heron's_formula
Mathematical treatise by Euclid
geometry and triangle congruence (1–26), parallel lines (27–34), the area of triangles and parallelograms (35–45), and the Pythagorean theorem and its
Euclid's_Elements
Study of triangles in other spaces than the Euclidean plane
MR 1161284, S2CID 123684622 Masala, G. (1999), "Regular triangles and isoclinic triangles in the Grassmann manifolds G2(RN)", Rendiconti del Seminario
Generalized_trigonometry
Texts belonging to the Śrauta ritual
statements of the Pythagorean theorem, both in the case of an isosceles right triangle and in the general case, as well as lists of Pythagorean triples. In
Shulba_Sutras
Shape with four equal sides and angles
permute the eight isosceles triangles between the half-edges and the square's center (which stays in place); any of these triangles can be taken as the fundamental
Square
90° angle (π/2 radians)
addition to equal-length sides. The Pythagorean theorem states how to determine when a triangle is a right triangle. In Unicode, the symbol for a right
Right_angle
Ancient Chinese mathematics text
side of the right triangle while knowing the other two. Gou Gu integer is precisely the finding of some significant integer Pythagorean numbers, including
The Nine Chapters on the Mathematical Art
The_Nine_Chapters_on_the_Mathematical_Art
Theorem in geometry
of the Pythagorean proposition, where squares are placed on the edges of triangles, was to place equilateral triangles on the edges of triangles: could
Napoleon's_theorem
On integer partitions from monotonic functions
JSTOR 40148160, MR 1189138 Wild, Roy E. (1955), "On the number of primitive Pythagorean triangles with area less than n", Pacific Journal of Mathematics, 5: 85–91
Lambek–Moser_theorem
Babylonian clay tablet on mathematics
the last part of the text, the solution is proved correct using the Pythagorean theorem. The steps of the solution are believed to represent cut-and-paste
IM_67118
Mathematical puzzle
or numerically for the wall heights A and B, and the Pythagorean theorem on one of the triangles can be used to solve for the width w. The problem may
Crossed_ladders_problem
Fundamental trigonometric functions
choice of a right triangle containing an angle of measure α {\displaystyle \alpha } . However, this is not the case as all such triangles are similar, and
Sine_and_cosine
Ancient Chinese proof of the Pythagorean theorem
right triangle to demonstrate the Pythagorean theorem. However the Chinese people seem to have generalized its conclusion to all right triangles. The hsuan
Xuan_tu
Rectangle constructed from 4 right-angled triangles
the original triangles' top corners creates a 45°–45°–90° triangle between the two, with sides of lengths 2, 2, and (by the Pythagorean theorem) 2 2 {\displaystyle
Ailles_rectangle
English courtier and diplomat
a demonstration, using his method of descent, that the area of a Pythagorean triangle cannot be a square. His Discourse Concerning the Vegetation of Plants
Kenelm_Digby
Historical development of geometry
of the Pythagorean theorem for all triangles, before which proofs only existed for the theorem for the special cases of a special right triangle. A 2007
History_of_geometry
Plane fractal constructed from squares
of touching squares encloses a right triangle, in a configuration traditionally used to depict the Pythagorean theorem. If the largest square has a size
Pythagoras_tree_(fractal)
main building [uk] and a negative of its seal in the background "Pythagorean triangle" as seen by Skovoroda, often interpreted as a Masonic symbol 15 September
Banknotes of the Ukrainian hryvnia
Banknotes_of_the_Ukrainian_hryvnia
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
Girl/Female
Celebrity, Gujarati, Hindu, Indian, Kannada, Sanskrit, Traditional
Goddess of Melody; Master of Melodic Modes
Boy/Male
German Greek Swedish
noble.
Girl/Female
Hindu
Early morning fragrance or entertaining companion or wind, Enchanting
Biblical
my light, my fire
Surname or Lastname
English (mainly central)
English (mainly central) : topographic name for someone who lived where holly trees grew, from Middle English holi(n)s, plural of holin, holi(e) (Old English hole(g)n).
Girl/Female
Hindu
Happy, Joyous
Male
Danish
, cheerful.
Girl/Female
Tamil
Aswini | à®…à®·à¯à®µà®¿à®¨à¯€
It is a name of a star
Girl/Female
Tamil
Nushka | நà¯à®‚à®·à¯à®•à®¾
Precious possession
Boy/Male
Indian
A literary person, Cultured, Civilized
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
PYTHAGOREAN TRIANGLES
a.
See Pythagorean, a.
n.
A solid related to a tetrahedron, and contained under twelve equal triangles.
n.
A solid bounded by eight faces. The regular octahedron is contained by eight equal equilateral triangles.
n.
The series or network of triangles into which the face of a country, or any portion of it, is divided in a trigonometrical survey; the operation of measuring the elements necessary to determine the triangles into which the country to be surveyed is supposed to be divided, and thus to fix the positions and distances of the several points connected by them.
a.
Of or pertaining to Pythagoras (a Greek philosopher, born about 582 b. c.), or his philosophy.
n.
A follower of Pythagoras; one of the school of philosophers founded by Pythagoras.
n.
An instrument for constructing triangles in marine surveying, etc.
n.
To occupy the same place in space, as two equal triangles, when placed one on the other.
n.
A solid figure contained by a plane rectilineal figure as base and several triangles which have a common vertex and whose bases are sides of the base.
n.
A solid figure inclosed or bounded by four triangles.
n.
A figure composed of two equilateral triangles intersecting so as to form a six-pointed star, -- used in early ornamental art, and also with superstitious import by the astrologers and mystics of the Middle Ages.
v. i.
To speculate after the manner of Pythagoras.
n.
The doctrines taught by Pythagoras.
n.
The doctrines of Pythagoras or the Pythagoreans.
n.
In Gothic vaulting, one of the primary members of the vault. These are strong arches, meeting and crossing one another, dividing the whole space into triangles, which are then filled by vaulted construction of lighter material. Hence, an imitation of one of these in wood, plaster, or the like.
a.
Having, or being in, a contrary order; -- said of a section of an oblique cone having a circular base made by a plane not parallel to the base, but so inclined to the axis that the section is a circle; applied also to two similar triangles when so placed as to have a common angle at the vertex, the opposite sides not being parallel.
n.
A wedge-shaped crystal bounded by four equal isosceles triangles. It is the hemihedral form of a square pyramid.
a.
Having the two things that are conjugate parts of the same figure; as, self-conjugate triangles.
n.
That branch of mathematics which treats of the relations of the sides and angles of triangles, which the methods of deducing from certain given parts other required parts, and also of the general relations which exist between the trigonometrical functions of arcs or angles.
v. t.
To divide into triangles; specifically, to survey by means of a series of triangles properly laid down and measured.