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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.
If the short side of a Kepler triangle has length , the other sides will have lengths and . The area can be calculated by the standard formula for the area of right triangles (half the product of the two short sides) as s 2 2 φ {\displaystyle {\tfrac {s^{2}}{2}}{\sqrt {\varphi }}} .
Possible geometric basis for a solution of IM 67118. Solid lines of the figure show stage 1; dashed lines and shading show stage 2. The central square has side b − a. The light gray region is the gnomon of area A = ab. The dark gray square (of side (b − a)/2) completes the gnomon to a square of side (b + a)/2.
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).
As a consequence of the Pythagorean theorem, the hypotenuse is the longest side of any right triangle; that is, the hypotenuse is longer than either of the triangle's legs. For example, given the length of the legs a = 5 and b = 12, then the sum of the legs squared is (5 × 5) + (12 × 12) = 169, the square of the hypotenuse.
If the apex of the pyramid is directly above the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral. It is called an equilateral square pyramid, an example of a Johnson solid.
Chapter 10 describes Pythagorean triangles with a side or area that is a square or cube, connecting this problem to Fermat's Last Theorem. After a chapter on Heronian triangles , Chapter 12 returns to this theme, discussing triangles whose hypotenuse and sum of sides are squares.
The Pythagorean theorem: The sum of the areas of the two squares on the legs (a and b) equals the area of the square on the hypotenuse (c). Although Pythagoras is most famous today for his alleged mathematical discoveries, [132] [207] classical historians dispute whether he himself ever actually made any significant contributions to the field.