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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.
The Bride's chair proof of the Pythagorean theorem, that is, the proof of the Pythagorean theorem based on the Bride's Chair diagram, is given below. 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 ...
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).
Two New Orleans high school students have proven the Pythagorean Theorem using trigonometry without relying on circular reasoning. That should be impossible.
There are many ways to prove Heron's formula, for example using trigonometry as below, or the incenter and one excircle of the triangle, [7] or as a special case of De Gua's theorem (for the particular case of acute triangles), [8] or as a special case of Brahmagupta's formula (for the case of a degenerate cyclic quadrilateral).
The method of chapter 7 was not found in Europe until the 13th century, and the method of chapter 8 uses Gaussian elimination before Carl Friedrich Gauss (1777–1855). [3] There is also the mathematical proof given in the treatise for the Pythagorean theorem. [4]
The book provided illustrated proof for the Pythagorean theorem, [31] contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon, the circle and square, as well as measurements of heights and distances. [32]
He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. [11] Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, [12] though the statement of the theorem has a long history.