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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 second chapter concerns plane geometry, and includes material on the golden ratio, [5] the Pythagorean theorem, [6] dissection problems, the mathematics of paper folding, tessellations, and plane curves, which are constructed by stitching, by graphical methods, and by mechanical devices. [1]
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).
Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as the mathematical model of space. Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority.
The Pythagorean theorem fails in elliptic geometry. In the 90°–90°–90° triangle described above, all three sides have the same length, and consequently do not satisfy + =. The Pythagorean result is recovered in the limit of small triangles.
By the Pythagorean theorem we have b 2 = h 2 + d 2 and a 2 = h 2 + (c − d) 2 according to the figure at the right. Subtracting these yields a 2 − b 2 = c 2 − 2cd. This equation allows us to express d in terms of the sides of the triangle: = + +.
Pythagorean fields can be used to construct models for some of Hilbert's axioms for geometry (Iyanaga & Kawada 1980, 163 C).The coordinate geometry given by for a Pythagorean field satisfies many of Hilbert's axioms, such as the incidence axioms, the congruence axioms and the axioms of parallels.