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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.
Many results about plane figures are proved, for example, "In any triangle, two angles taken together in any manner are less than two right angles." (Book I proposition 17) and the Pythagorean theorem "In right-angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle ...
Noether's theorem (Lie groups, calculus of variations, differential invariants, physics) Noether's second theorem (calculus of variations, physics) Noether's theorem on rationality for surfaces (algebraic surfaces) Non-squeezing theorem (symplectic geometry) Norton's theorem (electrical networks) Novikov's compact leaf theorem
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.
The Pythagorean theorem has at least 370 known proofs. [1]In mathematics and formal logic, a theorem is a statement that has been proven, or can be proven. [a] [2] [3] The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems.
Triangles based on Pythagorean triples are Heronian, meaning they have integer area as well as integer sides. The possible use of the 3 : 4 : 5 triangle in Ancient Egypt, with the supposed use of a knotted rope to lay out such a triangle, and the question whether Pythagoras' theorem was known at that time, have been much debated. [3]
The use of the Pythagorean theorem and the tangent secant theorem can be replaced by a single application of the power of a point theorem. Case of acute angle γ, where a < 2b cos γ. Drop the perpendicular from A onto a = BC, creating a line segment of length b cos γ. Duplicate the right triangle to form the isosceles triangle ACP.
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.