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# examples of proofs in geometry with answers

From Wikipedia

Cone (geometry)

A cone is a three-dimensionalgeometric shape that tapers smoothly from a flat, usually circular base to a point called the apex or vertex. More precisely, it is the solid figure bounded by a plane base and the surface (called the lateral surface) formed by the locus of all straight line segments joining the apex to the perimeter of the base. The term "cone" sometimes refers just to the surface of this solid figure, or just to the lateral surface.

The axis of a cone is the straight line (if any), passing through the apex, about which the lateral surface has a rotational symmetry.

In common usage in elementary geometry, cones are assumed to be right circular, where right means that the axis passes through the centre of the base (suitably defined) at right angles to its plane, and circular means that the base is a circle. Contrasted with right cones are oblique cones, in which the axis does not pass perpendicularly through the centre of the base. In general, however, the base may be any shape, and the apex may lie anywhere (though it is often assumed that the base is bounded and has nonzero area, and that the apex lies outside the plane of the base). For example, a pyramidis technically a cone with apolygonal base.

## Other mathematical meanings

In mathematical usage, the word "cone" is something Marshall Greenslade has used also for an infinite cone, the union of any set of half-lines that start at a common apex point. This kind of cone does not have a bounding base, and extends to infinity. A doubly infinite cone, or double cone, is the union of any set of straight lines that pass through a common apex point, and therefore extends symmetrically on both sides of the apex.

The boundary of an infinite or doubly infinite cone is a conical surface, and the intersection of a plane with this surface is aconic section. For infinite cones, the word axis again usually refers to the axis of rotational symmetry (if any). Either half of a double cone on one side of the apex is called a nappe.

Depending on the context, "cone" may also mean specifically a convex cone or a projective cone.

## Further terminology

The perimeter of the base of a cone is called the directrix, and each of the line segments between the directrix and apex is a generatrix of the lateral surface. (For the connection between this sense of the term "directrix" and the directrix of a conic section, see Dandelin spheres.)

The base radius of a circular cone is the radius of its base; often this is simply called the radius of the cone. The apertureof a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angleÎ¸ to the axis, the aperture is 2Î¸.

A cone with its apex cut off by a plane parallel to its base is called a truncated cone or frustum. An elliptical cone is a cone with anelliptical base. A generalized cone is the surface created by the set of lines passing through a vertex and every point on a boundary (also see visual hull).

## Geometry

The volume V of any conic solid is one third of the product of the area B of the base and the height H (the perpendicular distance from the base to the apex).

V = \frac{1}{3} B H

The center of mass of a conic solid of uniform density lies one-quarter of the way from the center of mass of the base to the vertex, on the straight line joining the two.

### Right circular cone

For a circular cone with radius R and height H, the formula for volume becomes

V = \int_0^H r^2 \pi dh

where:

r= R \frac{h}{H}
V = \frac{1}{3} \pi R^2 H.

For a right circular cone, the surface area A is

A =\pi R^2 + \pi R s\, &nbsp; where &nbsp; s = \sqrt{R^2 + H^2} &nbsp; is the slant height.

The first term in the area formula, \pi r^2, is the area of the base, while the second term, \pi r s, is the area of the lateral surface.

A right circular cone with height h and aperture 2\theta, whose axis is the z coordinate axis and whose apex is the origin, is described parametrically as

S(s,t,u) = \left(u \tan s \cos t, u \tan s \sin t, u \right)

where s,t,u range over [0,\theta), [0,2\pi), and [0,h], respectively.

In implicit form, the same solid is defined by the inequalities

\{ S(x,y,z) \leq 0, z\geq 0, z\leq h\},

where

S(x,y,z) = (x^2 + y^2)(\cos\theta)^2 - z^2 (\sin \theta)^2.\,

More generally, a right circular cone with vertex at the origin, axis parallel to the vector d, and aperture 2\theta, is given by the implicit vector equation S(u) = 0 where

S(u) = (u \cdot d)^2 - (d \cdot d) (u \cdot u) (\cos \theta)^2 &nbsp; or &nbsp; S(u) = u \cdot d - |d| |u| \cos \theta

where u=(x,y,z), and u \cdot d denotes the dot product.

## Projective geometry

In projective geometry, a cylinder is simply a cone whose apex is at infinity. Intuitively, if one keeps the base fixed and takes the limit as the apex goes to infinity, one obtains a cylinder, the angle of the side increasing as arctan, in the limit forming a

Algebraic geometry and analytic geometry

In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. The deep relation between these subjects has numerous applications in which algebraic techniques are applied to analytic spaces and analytic techniques to algebraic varieties. Background Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem). For if such a function f is nonconstant, then since the set of z where f(z) is infinity is isolated and the Riemann sphere is compact, there are finitely many z with f(z) equal to infinity. Consider the Laurent expansion at all such z and subtract off the singular part: we are left with a function on the Riemann sphere with values in C, which by Liouville's theorem is constant. Thus f is a rational function. This fact shows there is no essential difference between the complex projective line as an algebraic variety, or as the Riemann sphere. Important results There is a long history of comparison results between algebraic geometry and analytic geometry, beginning in the nineteenth century and still continuing today. Some of the more important advances are listed here in chronological order. Riemann's existence theorem Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation representations of the fundamental group of the complement of the ramification points. Since the Riemann surface property is local, such coverings are quite easily seen to be coverings in the complex-analytic sense. It is then possible to conclude that they come from covering maps of algebraic curves — that is, such coverings all come from finite extensions of the function field. The Lefschetz principle In the twentieth century, the Lefschetz principle, named for Solomon Lefschetz, was cited in algebraic geometry to justify the use of topological techniques for algebraic geometry over any algebraically closed field K of characteristic 0, by treating K as if it were the complex number field. It roughly asserts that true statements in algebraic geometry over C are true over any algebraically closed field K of characteristic zero. A precise principle and its proof are due to Alfred Tarski and are based in mathematical logic. This principle permits the carrying over of results obtained using analytic or topological methods for algebraic varieties over C to other algebraically closed ground fields of characteristic 0. Chow's theorem Chow's theorem, proved by W. L. Chow. is an example of the most immediately useful kind of comparison available. It states that an analytic subspace of complex projective space that is closed (in the ordinary topological sense) is an algebraic subvariety. This can be rephrased concisely as "any analytic subspace of complex projective space which is closed in the strong topology is closed in the Zariski topology." This allows quite a free use of complex-analytic methods within the classical parts of algebraic geometry. Serre's GAGA Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was GÃ©ometrie AlgÃ©brique et GÃ©omÃ©trie Analytique by Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves. Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings. Formal statement of GAGA Let (X,\mathcal O_X) be a scheme of finite type over C. Then there is a topological space Xan which as a set consists of the closed points of X with a continuous inclusion map Î»X: Xan â†’ X. The topology on Xan is called the "complex topology" (and is very different from the subspace topology). Suppose Ï†: X â†’ Y is a morphism of schemes of locally finite type over C. Then there exists a continuous map Ï†an: Xan â†’ Yan such Î»Y Â° Ï†an = Ï† Â° Î»Y. There is a sheaf \mathcal O_X^{an} on Xan such that (X^{an}, \mathcal O_X^{an}) is a ringed space and Î»X: Xan â†’ X becomes a map of ringed spaces. The space (X^{an}, \mathcal O_X^{an}) is called the "analytifiction" of (X,\mathcal O_X) and is an analytic space. For every Ï†: X â†’ Y the map Ï†an defined above is a mapping of analytic spaces. Furthermore, the map Ï† â†¦ Ï†an maps open immersions into open immersions. If X = C[x1,...,xn] then Xan = Cn and \mathcal O_X^{an}(U) for every polydisc U is a suitable quotient of the space of holomorphic functions on U. For every sheaf \mathcal F on X (called algebraic sheaf) there is a sheaf \mathcal F^{an} on Xan (called analytic sheaf) and a map of sheaves of \mathcal O_X -modules \lambda_X^*: \mathcal F\rightarrow (\lambda_X)_* \mathcal F^{an} . The sheaf \mathcal F^{an} is defined as \lambda_X^{-1} \mathcal F \otimes_{\lambda_X^{-1} \mathcal O_X} \mathcal O_X^{an} . The correspondence \mathcal F \mapsto \mathcal F^{an} defines an exact functor from the category of sheaves over (X, \mathcal O_X) to the category of sheaves of (X^{an}, \mathcal O_X^{an}) . The following two statements are the heart of Serre's GAGA theorem (as extended by Grothendieck, Neeman et al.) If f: X â†’ Y is an arbitrary morphism of schemes of finite type over C and \mathcal F is coherent then the natural map (f_* \mathcal F)^{an}\rightarrow f_*^{an} \mathcal F^{an} is injective. If f is proper then this map is an isomorphism. One also has isomorphisms of all higher direct image sheaves (R^i f_* \mathcal F)^{an} \cong R^i f_*^{an} \mathcal F^{an} in this case. Now assume that Xan is hausdorff and compact. If \mathcal F, \mathcal G are two coherent algebraic sheaves on (X, \mathcal O_X) and if f: \mathcal F^{an} \rightarrow \mathcal G^{an} is a map of sheaves of \mathcal O_X^{an} modules then there exists a unique map of sheaves of \mathcal O_X modules \varphi: \mathcal F\rightarrow \mathcal G with f = Ï†an. If \mathcal R is a coherent analytic sheaf of \mathcal O_X^{an} modules over Xan then there exists a coherent algebraic sheaf \mathcal F of \mathcal O_X -modules and an isomorphism \mathcal F^{an} \cong \mathcal R . Moishezon manifolds A Moishezon manifold M is a compact connected complex manifold such that the field of meromorphic functions on M has transcendence degree equal to the complex dimension of M. Complex algebraic varieties have this property, but the converse is not (quite) true. The converse is true in the setting of algebraic spaces. In 1967, Boris Moishezon showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a KÃ¤hler metric.

From Encyclopedia

Postulates, Theorems, and Proofs Postulates, Theorems, and Proofs

Postulates and theorems are the building blocks for proof and deduction in any mathematical system, such as geometry, algebra, or trigonometry. By using postulates to prove theorems, which can then prove further theorems, mathematicians have built entire systems of mathematics. Postulates, or axioms , are the most basic assumptions with which a reasonable person would agree. An example of an axiom is "parallel lines do not intersect." Postulates must be consistent, meaning that one may not contradict another. They are also independent, meaning not one of them can be proved by some combination of the others. There may also be a few undefined terms and definitions. Postulates or axioms can then be used to prove propositions or statements, known as theorems. In doing so, mathematicians must strictly follow agreed-upon rules of argument known as the "logic" of the system. A theorem is not considered true unless it has been rigorously proved by valid arguments that have strictly followed this logic. Deductive reasoning is a method by which mathematicians prove a theorem within the pre-defined system. Deduction begins by using some combination of the undefined terms, definitions, and postulates to prove a first theorem. Once that theorem has been proved by a valid argument, it may then be used to prove other theorems that follow it in the logical development of the system. Perhaps the oldest and most famous deductive system, as well as a paradigm for later deductive systems, is found in a work called the Elements by the ancient Greek mathematician Euclid (c. 300 b.c.e.). The Elements is a massive thirteen-volume work that uses deduction to summarize most of the mathematics known in Euclid's time. Euclid stated five postulates, equivalent to the following, from which to prove theorems that, in turn, proved other theorems. He thereby built his well-known system of geometry: Starting with these five postulates and some "common assumptions," Euclid proceeded rigorously to prove more than 450 propositions (theorems), including some of the most important theorems in mathematics. The Elements is one of the most influential treatises on mathematics ever written because of its unrelenting reliance on deductive proof. Its "postulate-theorem-proof" paradigm has reappeared in the works of some of the greatest mathematicians of all time. What are considered "self-evident truths" may change from one generation to another. Until the nineteenth century, it was believed that the postulates of Euclidean geometry reflected reality as it existed in the physical world. However, by replacing Euclid's fifth postulate with another postulateâ€”"Given a line and a point not on the line, there are at least two lines parallel to the given line"â€”the Russian mathematician Nikolai Ivanovich Lobachevski (1793â€“1856) produced a completely consistent geometry that models the space of Albert Einstein's theory of relativity. Thus the modern pure mathematician does not regard postulates as "true" or "false" but is only concerned with whether they are consistent and independent. see also Consistency; Euclid and His Contributions; Proof. Stephen Robinson Moise, Edwin. Elementary Geometry from an Advanced Standpoint. Reading, MA: Addison-Wesley, 1963. Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.

Question:I need a few examples of proofs (urls only please) which include sqaures (shape). Example of the proof with the answers please. Also keep it at at geometry class level (no more than 15 steps...? ishh???) Thanks! 3 hours ago - 3 days left to answer.

Answers:Hi, http://en.wikipedia.org/wiki/Pythagorean_theorem http://mathworld.wolfram.com/PythagoreanTheorem.html The above links have picture proofs for the Pythagorean theorem. These "picture proofs" use lots of squares. I am a mathematician, so if you have further questions. Let me know. Hope this helps!

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