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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between 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.
A triangle whose side lengths are a Pythagorean triple is a right triangle and called a Pythagorean triangle. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not.
Since at least the first century BC, Pythagoras has commonly been given credit for discovering the Pythagorean theorem, [212] [213] a theorem in geometry that states that "in a right-angled triangle the square of the hypotenuse is equal [to the sum of] the squares of the two other sides" [214] —that is, + =.
Garfield's proof of the Pythagorean theorem is an original proof the Pythagorean theorem discovered by James A. Garfield (November 19, 1831 – September 19, 1881), the 20th president of the United States. The proof appeared in print in the New-England Journal of Education (Vol. 3, No.14, April 1, 1876).
Early-Pythagorean philosophers proved simple geometrical theorems, including "the sum of the angles of a triangle equals two right angles". Pythagoreans also came up with three of the five platonic solids : the tetrahedron , the cube and the dodecahedron .
A tree of primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean triple is represented by exactly one node. In two of these trees, Berggren's tree and Price's tree, the root of the tree is the triple (3,4,5), and each node has exactly three children ...
There is a method to construct all Pythagorean triples that contain a given positive integer x as one of the legs of the right-angled triangle associated with the triple. It means finding all right triangles whose sides have integer measures, with one leg predetermined as a given cathetus. [13] The formulas read as follows.
In geometry, a Bride's Chair is an illustration of the Pythagorean theorem. [1] The figure appears in Proposition 47 of Book I of Euclid's Elements . [ 2 ] It is also known by several other names, such as the Franciscan's cowl , peacock's tail , windmill , Pythagorean pants , Figure of the Bride , theorem of the married women , and chase of the ...