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In geometry, the hinge theorem (sometimes called the open mouth theorem) states that if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second triangle. [1 ...
When k = 1, Babbage's theorem implies that it holds for n = p 2 for p an odd prime, while Wolstenholme's theorem implies that it holds for n = p 3 for p > 3, and it holds for n = p 4 if p is a Wolstenholme prime. When k = 2, it holds for n = p 2 if p is a Wolstenholme prime. These three numbers, 4 = 2 2, 8 = 2 3, and 27 = 3 3 are not held for ...
The parameters most commonly appearing in triangle inequalities are: the side lengths a, b, and c;; the semiperimeter s = (a + b + c) / 2 (half the perimeter p);; the angle measures A, B, and C of the angles of the vertices opposite the respective sides a, b, and c (with the vertices denoted with the same symbols as their angle measures);
It was originally given by Jürgen Moser in 1965 to check when two volume forms are equivalent, [1] but its main applications are in symplectic geometry. It is the standard argument for the modern proof of Darboux's theorem, as well as for the proof of Darboux-Weinstein theorem [2] and other normal form results. [2] [3] [4]
The converse of the triangle inequality theorem is also true: if three real numbers are such that each is less than the sum of the others, then there exists a triangle with these numbers as its side lengths and with positive area; and if one number equals the sum of the other two, there exists a degenerate triangle (that is, with zero area ...
Graham's number is an immense number that arose as an upper bound on the answer of a problem in the mathematical field of Ramsey theory.It is much larger than many other large numbers such as Skewes's number and Moser's number, both of which are in turn much larger than a googolplex.
In 1999, a theorem due to Haga provided constructions used to divide the side of a square into rational fractions. [19] [20] In late 2001 and early 2002, Britney Gallivan proved the minimum length of paper necessary to fold it in half a certain number of times and folded a 4,000-foot-long (1,200 m) piece of toilet paper twelve times. [21] [22]
This case of the theorem is still provable by Π 1 1-CA 0, but by adding a "gap condition" [3] to the definition of the order on trees above, he found a natural variation of the theorem unprovable in this system. [4] [5] Much later, the Robertson–Seymour theorem would give another theorem unprovable by Π 1 1-CA 0.