Pythagoras theorem wikipedia

Consider the triangle shown below. This figure is clearly a squaresince all pythagoras theorem wikipedia angles are right anglesand the lines connecting the corners are easily seen to be straight.

Garfield's proof of the Pythagorean theorem is an original proof the Pythagorean theorem invented by James A. Garfield November 19, — September 19, , the 20th president of the United States. He assumed the office of President on March 4, , and served in that position only for a brief period up to September 19, The proof is nontrivial and, according to the historian of mathematics, William Dunham , "Garfield's is really a very clever proof. The theorem is proved by computing the area of this trapezoid in two different ways.

Pythagoras theorem wikipedia

Such a triple is commonly written a , b , c , and a well-known example is 3, 4, 5. If a , b , c is a Pythagorean triple, then so is ka , kb , kc for any positive integer k. A primitive Pythagorean triple is one in which a , b and c are coprime that is, they have no common divisor larger than 1. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle and is a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton , a Babylonian clay tablet from about BC, written in a sexagesimal number system. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. Other small Pythagorean triples such as 6, 8, 10 are not listed because they are not primitive; for instance 6, 8, 10 is a multiple of 3, 4, 5. Each of these points with their multiples forms a radiating line in the scatter plot to the right.

It can be proved using the law of cosines or as follows:.

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]. This theorem should not be confused with proposition 48 in book 1 of Euclid 's Elements , the converse of the Pythagorean theorem, which states that if the square on one side of a triangle is equal to the sum of the squares on the other two sides then the other two sides contain a right angle. Using the Pythagorean theorem ,. The cruciform curve or cross curve is a quartic plane curve given by the equation. Substituting x with AC and y with BC gives. Inverse-Pythagorean triples can be generated using integer parameters t and u as follows. If two identical lamps are placed at A and B , the theorem and the inverse-square law imply that the light intensity at C is the same as when a single lamp is placed at D.

These means were studied with proportions by Pythagoreans and later generations of Greek mathematicians [1] because of their importance in geometry and music. This is a generalization of the inequality of arithmetic and geometric means and a special case of an inequality for generalized means. The study of the Pythagorean means is closely related to the study of majorization and Schur-convex functions. The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur-concave, while the arithmetic mean is a linear function of its arguments and hence is both concave and convex. Almost everything that we know about the Pythagorean means came from arithmetic handbooks written in the first and second century. There are three means in music: one is arithmetic, second is the geometric, third is sub-contrary, which they call harmonic. The mean is arithmetic when three terms are in proportion such that the excess by which the first exceeds the second is that by which the second exceeds the third. In this proportion it turns out that the interval of the greater terms is less, but that of the lesser terms greater. The mean is the geometric when they are such that as the first is to the second, so the second is to the third. Of these terms the greater and the lesser have the interval between them equal.

Pythagoras theorem wikipedia

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CRC Press. According to Thomas L. A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem , named for Jean Paul de Gua de Malves : If a tetrahedron has a right angle corner like a corner of a cube , then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. For such a triple, either a or b is even, and the other is odd; from this, it follows that c is also odd. Since the sum F k , m of k consecutive squares beginning with m 2 is given by the formula, [41]. The theorem is proved by computing the area of this trapezoid in two different ways. Proof using similar triangles [ change change source ] We can get another proof of the Pythagorean theorem by using similar triangles. Since both triangles' sides are the same lengths a , b and c , the triangles are congruent and must have the same angles. This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right angle in the triangles, forming a square in the center whose sides are length c. Pythagoras of Samos ". As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed.

In mathematics , a theorem is a statement that has been proved , or can be proved. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo—Fraenkel set theory with the axiom of choice ZFC , or of a less powerful theory, such as Peano arithmetic. Moreover, many authors qualify as theorems only the most important results, and use the terms lemma , proposition and corollary for less important theorems.

Written c. CRC concise encyclopedia of mathematics 2nd ed. Tools Tools. The theorem is proved by computing the area of this trapezoid in two different ways. Leucippus Democritus. The freemasons deliberately modeled their society on the community founded by Pythagoras at Croton. Germany Trove. The triple generated by Euclid 's formula is primitive if and only if m and n are coprime and one of them is even. Read Edit View history. The Transcendentalists read the ancient Lives of Pythagoras as guides on how to live a model life. At the same time the triangle lengths are measured as shown, with the hypotenuse of length y , the side AC of length x and the side AB of length a , as seen in the lower diagram part. Although Pythagoras is most famous today for his alleged mathematical discoveries, [] [] classical historians dispute whether he himself ever actually made any significant contributions to the field. Samuel Weiser, Inc.