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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 Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem.In other words, it is a function that satisfies a particular intermediate-value property — on any interval (,), the function takes every value between () and () — but is not continuous.
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.
The product of this form of 1 / 3 with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.
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.
In mathematics, the ratio test is a test (or "criterion") for the convergence of a series =, where each term is a real or complex number and a n is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.