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Τι (ποιος) είναι pythagorean$65906$ - ορισμός

THREE POSITIVE INTEGERS, THE SQUARES OF TWO OF WHICH SUM TO THE SQUARE OF THE THIRD
Pythagorean triangle; Pythagorean triplet; Pythagorean triples; Platonic sequence; Pythagorean triad; Pythagorean Triple; Pythagorean triplets; Pythagorean right triangle; Pythagorian triple; Euclid's formula; Primitive Pythagorean triple; Primitive Pythagorean Triple; Pythagorean n-tuple; Primitive Pythagorean triangle; Decomposable Pythagorean triangle
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Pythagorean triple         
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is .
https://en.wikipedia.org/wiki/Pythagorean_triple
Pythagorean theorem         
  • Rearrangement proof of the Pythagorean theorem.<br>(The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always '''''c²'''''. And likewise, at all moments in time, the area is always '''''a²+b²'''''.)
  • Geometric proof of the Pythagorean theorem from the ''[[Zhoubi Suanjing]]''
  • The absolute value of a complex number ''z'' is the distance ''r'' from ''z'' to the origin.
  • '''a × b'''}} is normal to this plane.
  • Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof.
  • The [[spiral of Theodorus]]: A construction for line segments with lengths whose ratios are the square root of a positive integer
  • [[Hyperbolic triangle]]
  • Proof in Euclid's ''Elements''
  • Illustration including the new lines
  • Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF
  • (r<sub>2</sub>, θ<sub>2</sub>)}} in [[polar coordinates]] is given by the [[law of cosines]]. Interior angle Δθ = θ<sub>1</sub>−θ<sub>2</sub>.
  • Distance between infinitesimally separated points in [[Cartesian coordinates]] (top) and [[polar coordinates]] (bottom), as given by Pythagoras' theorem
  • Vectors involved in the parallelogram law
  • p=36}}.</ref>
  • Animation showing proof by rearrangement of four identical right triangles
  • Diagram for differential proof
  • Animation showing another proof by rearrangement
  • Pythagoras' theorem in three dimensions relates the diagonal AD to the three sides.
  • Diagram of the two algebraic proofs
  • A + B {{=}} blue}} area C
  • Generalization for regular pentagons
  • Pythagoras' theorem using similar right triangles
  • Construction for proof of parallelogram generalization
  • area {{=}} blue}} area
  • Proof using similar triangles
  • Proof using an elaborate rearrangement
  • A tetrahedron with outward facing right-angle corner
  • Spherical triangle
  • Similar right triangles showing sine and cosine of angle θ
  • publisher=Mathematical Association of America}} </ref> Lower panel: reflection of triangle CAD (top) to form triangle DAC, similar to triangle ABC (top).
  • Visual proof of the Pythagorean theorem by area-preserving shearing
RELATION IN EUCLIDEAN GEOMETRY AMONG THE THREE SIDES OF A RIGHT TRIANGLE
PythagoreanTheorem; Pythagorean Theorm; Pythagoras' Theorem; Pythagoras' theorem; Pythagorean Theorem; Pythagoras theorem; Pythagorean Theorum; Pythagoras' Theorem Proof; Pythagoras's Law; Pythagoras’ theorem; Pythagoras’ Theorem; Pythagorean theorum; Pythagorean equation; A² + b² = c²; A²+b²=c²; A2 + b2 = c2; A2+b2=c2; Pythagoras's theorem; Pythagorus's theorem; Pythagorus's theorum; Pythagoras's theorum; The Pythagorean theorem; Pythagorus' theorum; Pythagoras theory; Pythagorean Thm; Theorem of Pythagoras; 47th Problem of Euclid; Pythagorean theorem proof; Pythagoras Theorem; A^2+b^2=c^2; Pyth. thm; Pyth. theorem; Converse of Pyth. thm; Converse of Pyth. theorem; Pythagorean formula; Pythagorean theory; Gougu theorem; Gougu; Pythagoras' law; Gougu's Theorem; Generalizations of the Pythagorean theorem
In mathematics, the Pythagorean theorem, or Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides.
https://en.wikipedia.org/wiki/Pythagorean_theorem
Pythagoras's Theorem         
  • Rearrangement proof of the Pythagorean theorem.<br>(The area of the white space remains constant throughout the translation rearrangement of the triangles. At all moments in time, the area is always '''''c²'''''. And likewise, at all moments in time, the area is always '''''a²+b²'''''.)
  • Geometric proof of the Pythagorean theorem from the ''[[Zhoubi Suanjing]]''
  • The absolute value of a complex number ''z'' is the distance ''r'' from ''z'' to the origin.
  • '''a × b'''}} is normal to this plane.
  • Right triangle on the hypotenuse dissected into two similar right triangles on the legs, according to Einstein's proof.
  • The [[spiral of Theodorus]]: A construction for line segments with lengths whose ratios are the square root of a positive integer
  • [[Hyperbolic triangle]]
  • Proof in Euclid's ''Elements''
  • Illustration including the new lines
  • Showing the two congruent triangles of half the area of rectangle BDLK and square BAGF
  • (r<sub>2</sub>, θ<sub>2</sub>)}} in [[polar coordinates]] is given by the [[law of cosines]]. Interior angle Δθ = θ<sub>1</sub>−θ<sub>2</sub>.
  • Distance between infinitesimally separated points in [[Cartesian coordinates]] (top) and [[polar coordinates]] (bottom), as given by Pythagoras' theorem
  • Vectors involved in the parallelogram law
  • p=36}}.</ref>
  • Animation showing proof by rearrangement of four identical right triangles
  • Diagram for differential proof
  • Animation showing another proof by rearrangement
  • Pythagoras' theorem in three dimensions relates the diagonal AD to the three sides.
  • Diagram of the two algebraic proofs
  • A + B {{=}} blue}} area C
  • Generalization for regular pentagons
  • Pythagoras' theorem using similar right triangles
  • Construction for proof of parallelogram generalization
  • area {{=}} blue}} area
  • Proof using similar triangles
  • Proof using an elaborate rearrangement
  • A tetrahedron with outward facing right-angle corner
  • Spherical triangle
  • Similar right triangles showing sine and cosine of angle θ
  • publisher=Mathematical Association of America}} </ref> Lower panel: reflection of triangle CAD (top) to form triangle DAC, similar to triangle ABC (top).
  • Visual proof of the Pythagorean theorem by area-preserving shearing
RELATION IN EUCLIDEAN GEOMETRY AMONG THE THREE SIDES OF A RIGHT TRIANGLE
PythagoreanTheorem; Pythagorean Theorm; Pythagoras' Theorem; Pythagoras' theorem; Pythagorean Theorem; Pythagoras theorem; Pythagorean Theorum; Pythagoras' Theorem Proof; Pythagoras's Law; Pythagoras’ theorem; Pythagoras’ Theorem; Pythagorean theorum; Pythagorean equation; A² + b² = c²; A²+b²=c²; A2 + b2 = c2; A2+b2=c2; Pythagoras's theorem; Pythagorus's theorem; Pythagorus's theorum; Pythagoras's theorum; The Pythagorean theorem; Pythagorus' theorum; Pythagoras theory; Pythagorean Thm; Theorem of Pythagoras; 47th Problem of Euclid; Pythagorean theorem proof; Pythagoras Theorem; A^2+b^2=c^2; Pyth. thm; Pyth. theorem; Converse of Pyth. thm; Converse of Pyth. theorem; Pythagorean formula; Pythagorean theory; Gougu theorem; Gougu; Pythagoras' law; Gougu's Theorem; Generalizations of the Pythagorean theorem
<mathematics> The theorem of geometry, named after Pythagoras, of Samos, Ionia, stating that, for a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. I.e. if the longest side has length A and the other sides have lengths B and C (in any units), A^2 = B^2 + C^2 (2004-02-12)

Βικιπαίδεια

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a 2 + b 2 = c 2 {\displaystyle 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 do not form Pythagorean triples. For instance, the triangle with sides a = b = 1 {\displaystyle a=b=1} and c = 2 {\displaystyle c={\sqrt {2}}} is a right triangle, but ( 1 , 1 , 2 ) {\displaystyle (1,1,{\sqrt {2}})} is not a Pythagorean triple because 2 {\displaystyle {\sqrt {2}}} is not an integer. Moreover, 1 {\displaystyle 1} and 2 {\displaystyle {\sqrt {2}}} do not have an integer common multiple because 2 {\displaystyle {\sqrt {2}}} is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.

When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation.

https://en.wikipedia.org/wiki/Pythagorean triple
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