Crossbar Theorem
geometry, the crossbar theorem states that if ray AD is between ray AC and ray AB, then ray AD intersects line segment BC. This result is one of the deeper results in axiomatic plane geometry. It is often used in proofs to justify the statement that a line through a vertex of a triangle lying ''inside'' the triangle meets the side of the triangle opposite that vertex. This property was often used by Euclid in his proofs without explicit justification. Some modern treatments (not Euclid's) of the proof of the theorem that the base angles of an isosceles triangle are congruent start like this: Let ABC be a triangle with side AB congruent to side AC. ''Draw the angle bisector of angle A and let D be the point at which it meets side BC''. And so on. The justification for the existence of point D is the often unstated crossbar theorem. For this particular result, other proofs exist which do not require the use of the crossbar theorem. See also * Foundations of geometry * Jordan curv ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Crossbar Theorem Diagram
Crossbar may refer to: Structures * Latch (hardware), a post barring a door * Top tube of a bicycle frame * Crossbar, the horizontal member of various sports goals * Crossbar, a horizontal member of an electricity pylon Other * In electronics, crossbar switch, a switch connecting multiple inputs to multiple outputs in a matrix manner * In typeface anatomy, ''crossbar'' refers to some types of horizontal strokes * Crossbar (computer hardware manufacturer), a company manufacturing resistive random-access memory * Crossbar (film) ''Crossbar'' is a Canadian television film, which aired on CBC Television in 1979."New high reached by CBC's Crossbar". '' Ottawa Journal'', July 23, 1979. A fictionalized account of the career of Canadian amputee athlete Arnie Boldt, the film sta ...

Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a '' geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geom ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Line (mathematics)
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are One-dimensional space, one-dimensional objects, though they may exist in Two-dimensional Euclidean space, two, Three-dimensional space, three, or higher dimension spaces. The word ''line'' may also refer to a line segment in everyday life, which has two Point (geometry), points to denote its ends. Lines can be referred by two points that lay on it (e.g., \overleftrightarrow) or by a single letter (e.g., \ell). Euclid described a line as "breadthless length" which "lies evenly with respect to the points on itself"; he introduced several postulates as basic unprovable properties from which he constructed all of geometry, which is now called Euclidean geometry to avoid confusion with other geometries which have been introduced since the end of the 19th century (such as Non-Euclidean geometry, non-Euclidean, Projective geometry, projective and affine geometry). In modern mathematic ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Line Segment
In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as \overline). Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve). In real or complex vector spaces If ''V'' is a vector spac ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Foundations Of Geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play. Axiomatic systems Based on ancient Greek methods, an ''axiomatic system'' is a formal description of a way to establish the ''mathematical truth'' that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Jordan Curve Theorem
In topology, the Jordan curve theorem asserts that every '' Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterior points. Every continuous path connecting a point of one region to a point of the other intersects with the curve somewhere. While the theorem seems intuitively obvious, it takes some ingenuity to prove it by elementary means. ''"Although the JCT is one of the best known topological theorems, there are many, even among professional mathematicians, who have never read a proof of it."'' (). More transparent proofs rely on the mathematical machinery of algebraic topology, and these lead to generalizations to higher-dimensional spaces. The Jordan curve theorem is named after the mathematician Camille Jordan (1838–1922), who found its first proof. For decades, mathematicians generally thought that this proof was flawed and that the firs ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Euclidean Plane Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of g ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Theorems In Plane Geometry
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' a ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]