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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 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 ...
Diagram of Pythagoras Theorem. Observe that we have four right-angled triangles and a square packed into a larger square. Each of the triangles has sides a and b and hypotenuse c. The area of a square is defined as the square of the length of its sides. In this case, the area of the large square is (a + b) 2. However, the area of the large ...
The shapes would be a symbolic representation of the Pythagorean theorem, large enough to be seen from the Moon or Mars. Although credited in numerous sources as originating with Gauss, with exact details of the proposal set out, the specificity of detail, and even whether Gauss made the proposal, have been called into question.
The theorem for which the greatest number of different proofs have been discovered is possibly the Pythagorean theorem, with hundreds of proofs being published up to date. [3] Another theorem that has been proved in many different ways is the theorem of quadratic reciprocity .
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
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. 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 ...