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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 triangle angle sum theorem states that the sum of the three angles of any triangle, in this case angles α, β, and γ, will always equal 180 degrees. The Pythagorean theorem states that the sum of the areas of the two squares on the legs ( a and b ) of a right triangle equals the area of the square on the hypotenuse ( c ).
The Pythagorean theorem, and hence this length, can also be derived from the law of cosines in trigonometry. In a right triangle, the cosine of an angle is the ratio of the leg adjacent of the angle and the hypotenuse. For a right angle γ (gamma), where the adjacent leg equals 0, the cosine of γ also equals 0.
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 ...
Explicitly, they are defined below as functions of the known angle A, where a, b and h refer to the lengths of the sides in the accompanying figure. In the following definitions, the hypotenuse is the side opposite to the 90-degree angle in a right triangle; it is the longest side of the triangle and one of the two sides adjacent to angle A.
A geometer is a mathematician whose area of study is the historical aspects that define geometry, instead of the analytical geometric studies that becomes conducted from geometricians. Some notable geometers and their main fields of work, chronologically listed, are:
The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem.
Well known results such as Heron's formula for calculating the area of a triangle from its side lengths, or the inscribed angle theorem in the form that the angles subtended by a chord of a circle from other points on the circle are equal, are reformulated in terms of quadrance and spread, and thereby generalized to arbitrary fields of numbers.