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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.
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 three sides of a right triangle are related by the Pythagorean theorem, which in modern algebraic notation can be written a 2 + b 2 = c 2 , {\displaystyle a^{2}+b^{2}=c^{2},} where c {\displaystyle c} is the length of the hypotenuse (side opposite the right angle), and a {\displaystyle a} and b {\displaystyle b} are the lengths of the legs ...
The same year, Kepler wrote to Maestlin of the Kepler triangle, which combines the golden ratio with the Pythagorean theorem. Kepler said of these: Geometry has two great treasures: one is the theorem of Pythagoras, the other the division of a line into extreme and mean ratio.
[13] [14] These three ways of combining two numbers were all studied in ancient Greek mathematics, and are called the Pythagorean means. [15] Conversely, this can be taken as an alternative definition of the Kepler triangle: it is a right triangle whose edge lengths are the three Pythagorean means of some two numbers.
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 . [ 34 ]
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.
In 1988, The Theorem of Pythagoras was the first video produced by the series and reviews the Pythagorean theorem. [4] For all right triangles, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a 2 + b 2 = c 2). The theorem is named after Pythagoras of ancient Greece.