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Splitter (geometry)
In Euclidean geometry, a splitter is a line segment through one of the vertices of a triangle (that is, a cevian) that bisects the perimeter of the triangle. They are not to be confused with cleavers, which also bisect the perimeter but instead emanate from the midpoint of one of the triangle's sides. Properties The opposite endpoint of a splitter to the chosen triangle vertex lies at the point on the triangle's side where one of the excircles of the triangle is tangent to that side. This point is also called a splitting point of the triangle. It is additionally a vertex of the extouch triangle and one of the points where the Mandart inellipse is tangent to the triangle side. The three splitters concur at the Nagel point of the triangle, which is also called its splitting center. Generalization Some authors have used the term "splitter" in a more general sense, for any line segment that bisects the perimeter of the triangle. Other line segments of this type include the clea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean 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 geometry ... [...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 space o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (in plural form: vertices or vertexes) is a point where two or more curves, lines, or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object. In a polygon, a vertex is called " convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles); ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinear, determine a unique triangle and simultaneously, a unique plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of Euclid's Elements. The names used for modern classification are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cevian
In geometry, a cevian is a line that intersects both a triangle's vertex, and also the side that is opposite to that vertex. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name. Length Stewart's theorem The length of a cevian can be determined by Stewart's theorem: in the diagram, the cevian length is given by the formula :\,b^2m + c^2n = a(d^2 + mn). Less commonly, this is also represented (with some rearrangement) by the following mnemonic: :\underset = \!\!\!\!\!\! \underset Median If the cevian happens to be a median (thus bisecting a side), its length can be determined from the formula :\,m(b^2 + c^2) = a(d^2 + m^2) or :\,2(b^2 + c^2) = 4d^2 + a^2 since :\,a = 2m. Hence in this case :d= \frac\sqrt2 . Angle bisector If the cevian happens to be an angle bisector, its length obeys the formulas :\,(b + c)^2 = a^2 \l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bisection
In geometry, bisection is the division of something into two equal or congruent parts, usually by a line, which is then called a ''bisector''. The most often considered types of bisectors are the ''segment bisector'' (a line that passes through the midpoint of a given segment) and the ''angle bisector'' (a line that passes through the apex of an angle, that divides it into two equal angles). In three-dimensional space, bisection is usually done by a plane, also called the ''bisector'' or ''bisecting plane''. Perpendicular line segment bisector Definition *The perpendicular bisector of a line segment is a line, which meets the segment at its midpoint perpendicularly. The Horizontal intersector of a segment AB also has the property that each of its points X is equidistant from the segment's endpoints: (D)\quad , XA, = , XB, . The proof follows from and Pythagoras' theorem: :, XA, ^2=, XM, ^2+, MA, ^2=, XM, ^2+, MB, ^2=, XB, ^2 \; . Property (D) is usually used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Association Of America
The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure and applied mathematicians; computer scientists; statisticians; and many others in academia, government, business, and industry. The MAA was founded in 1915 and is headquartered at 1529 18th Street, Northwest in the Dupont Circle neighborhood of Washington, D.C. The organization publishes mathematics journals and books, including the ''American Mathematical Monthly'' (established in 1894 by Benjamin Finkel), the most widely read mathematics journal in the world according to records on JSTOR. Mission and Vision The mission of the MAA is to advance the understanding of mathematics and its impact on our world. We envision a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cleaver (geometry)
In geometry, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. They are not to be confused with splitters, which also bisect the perimeter, but with an endpoint on one of the triangle’s vertices instead of its sides. Construction Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle. The broken chord theorem of Archimedes provides another construction of the cleaver. Suppose the triangle to be bisected is , and that one endpoint of the cleaver is the midpoint of side . Form the circumcircle of and let be the midpoint of the arc of the circumcircle from to through . Then the other endpoint of the cleaver is the closest point of the triangle to , and can be found by dropping a perpendicular from to the longer of the two sides and . Related figures The three cleavers concur at a point, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimensional space whose endpoints are A = (a_1, a_2, \dots , a_n) and B = (b_1, b_2, \dots , b_n) is given by :\frac. That is, the ''i''th coordinate of the midpoint (''i'' = 1, 2, ..., ''n'') is :\frac 2. Construction Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction. The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It is more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excircle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incenter. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex , for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex , or the excenter of . Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extouch Triangle
In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle. Coordinates The vertices of the extouch triangle are given in trilinear coordinates by: :\begin T_A &= 0 : \csc^2 : \csc^2\\ T_B &= \csc^2 : 0 : \csc^2\\ T_C &= \csc^2 : \csc^2 : 0 \end or equivalently, where are the lengths of the sides opposite angles respectively, :\begin T_A &= 0 : \frac : \frac \\ T_B &= \frac : 0 : \frac \\ T_C &= \frac : \frac : 0. \end Related figures The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram. The Mandart inellipse is tangent to the sides of the reference triangle at the three vertices of the extouch triangle. Area The area of the extouch triangle, , is given by: :K_T= K\frac where and are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
