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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 ...
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);
The first of these quadratic inequalities requires r to range in the region beyond the value of the positive root of the quadratic equation r 2 + r − 1 = 0, i.e. r > φ − 1 where φ is the golden ratio. The second quadratic inequality requires r to range between 0 and the positive root of the quadratic equation r 2 − r − 1 = 0, i.e. 0 ...
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 pons asinorum in Oliver Byrne's edition of the Elements [1]. In geometry, the theorem that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (/ ˈ p ɒ n z ˌ æ s ɪ ˈ n ɔːr ə m / PONZ ass-ih-NOR-əm), Latin for "bridge of asses", or more descriptively as the isosceles triangle theorem.
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 ...
where A 1 and A 2 are the centers of the two circles and r 1 and r 2 are their radii. The power of a point arises in the special case that one of the radii is zero. If the two circles are orthogonal, the Darboux product vanishes. If the two circles intersect, then their Darboux product is
Casey's theorem and its converse can be used to prove a variety of statements in Euclidean geometry. For example, the shortest known proof [ 1 ] : 411 of Feuerbach's theorem uses the converse theorem.