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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.
In 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 geometry operations involving a diagram from which, it has been suggested, ancient Mesopotamians might, at an earlier time, have derived the Pythagorean theorem.
The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.
The celebrated Pythagorean theorem (book I, proposition 47) states that in any right triangle, 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 whose sides are the two legs (the two sides that meet at a right angle).
The proof has been severely criticized by the German philosopher Arthur Schopenhauer as being unnecessarily complicated, with construction lines drawn here and there and a long line of deductive steps. According to Schopenhauer, the proof is a "brilliant piece of perversity". [6] The basic idea of the Bride's Chair proof of the Pythagorean theorem
Solving the full version of the problem will be an even bigger triumph. You probably haven’t heard of the math subject Knot Theory . It’s taught in virtually no high schools, and few colleges.
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [8] or as a special case of De Gua's theorem (for the particular case of acute triangles), [9] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
The Pythagorean theorem is also ancient, but it could only take its central role in the measurement of distances after the invention of Cartesian coordinates by René Descartes in 1637. The distance formula itself was first published in 1731 by Alexis Clairaut . [ 33 ]