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A base for a neighborhood system of a point is a collection of open sets such that belongs to every member of , and any open set containing also contains a member of as a subset.

Geometry is the study of figures in a space of a given number of dimensions and of a given type. The most common types of geometry are plane geometry (dealing with objects like the point, line, circle, triangle, and polygon), solid geometry (dealing with objects like the line, sphere, and polyhedron), and spherical geometry (dealing with objects like the spherical triangle and spherical polygon). Geometry was part of the quadrivium taught in medieval universities.A mathematical pun notes that without geometry, life is pointless. An old children's joke asks, "What does an acorn say when it grows up?" and answers, "Geometry" ("gee, I'm a tree").Historically, the study of geometry proceeds from a small number of accepted truths (axioms or postulates), then builds up true statements using a systematic and rigorous step-by-step proof. However, there is much more to geometry than this relatively dry textbook..

Two geometric figures are said to be concentric if their centers coincide. The region between two concentric circles is called an annulus.The following table summarizes some concentric centralcircles.Kimberlingcentercirclesincenter Adams' circle, Conway circle, hexyl circle, incircletriangle centroid inner Napoleon circle, outer Napoleon circlecircumcenter circumcircle, second Brocard circle, second Droz-Farny circle, Stammler circleorthocenter anticomplementary circle, polar circle, first Droz-Farny circlenine-point center nine-point circle, Steiner circleSpieker center excircles radical circle, Spieker circleBrocard midpointGallatly circle, half-Moses circle, Moses circlemidpoint of the Brocard diameterBrocard circle, first Lemoine circle

According to Pólya, the Cartesian pattern is the resolution method for arithmetical or geometrical problems based on equations. The first step is to translate the question into one or more algebraic equalities, which express the relationship between the numerical data (the coefficients) and the quantities to be determined (the unknowns). This relationship can be described in text, or be depicted in a figure.The second step is to solve the equations.Normally, the quantity requested by the problem is only one, which permits us to reduce the procedure to a single equation, whose sides contain two different expressions of the same quantity. Consider, for example, a problem asking for one of the legs of a right triangle given that the length of this leg is half the length of the hypotenuse and that the other leg has length 1. If the unknown leg is denoted by and the hypotenuse by , then(1)as specified, and, moreover,(2)by the Pythagorean theorem...

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