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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 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 ...
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem [1] or the upside down Pythagorean theorem [2]) is as follows: [3] Let A, B be the endpoints of the hypotenuse of a right triangle ABC. Let D be the foot of a perpendicular dropped from C, the vertex of the right angle, to the hypotenuse. Then
Due to the Pythagorean theorem the number () has the simple geometric meanings shown in the diagram: For a point outside the circle () is the squared tangential distance | | of point to the circle . Points with equal power, isolines of Π ( P ) {\displaystyle \Pi (P)} , are circles concentric to circle c {\displaystyle c} .
Each problem includes an answer and a corresponding arithmetic algorithm. It is an important source on early Chinese cosmology , glossing the ancient idea of a round heaven over a square earth ( 天 圆 地 方 , tiānyuán dìfāng ) as similar to the round parasol suspended over some ancient Chinese chariots [ 10 ] or a Chinese chessboard ...
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin( α + β ) = sin α cos β + cos α sin ...
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 ...
Plimpton 322 is a Babylonian clay tablet, believed to have been written around 1800 BC, that contains a mathematical table written in cuneiform script.Each row of the table relates to a Pythagorean triple, that is, a triple of integers (,,) that satisfies the Pythagorean theorem, + =, the rule that equates the sum of the squares of the legs of a right triangle to the square of the hypotenuse.