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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] |
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 geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), 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 ... [...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 (geometry), 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 (geometry), chord (of that curve). In real or complex vector spa ... [...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 (geometry), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perimeter
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter. Formulas The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with \int_0^L \mathrms, where L is the length of the path and ds is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed piecewise smooth plane curve ... [...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 ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 In geometry, the Nagel point (named for Christian Heinrich von Nagel) is a triangle center, one of the points associated with a given triangle whose definition does not depend on the placement or scale of the triangle. It is the point of concur ... of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Bisector
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 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Considered the greatest mathematician of ancient history, and one of the greatest of all time,* * * * * * * * * * Archimedes anticipated modern calculus and analysis by applying the concept of the infinitely small and the method of exhaustion to derive and rigorously prove a range of geometrical theorems. These include the area of a circle, the surface area and volume of a sphere, the area of an ellipse, the area under a parabola, the volume of a segment of a paraboloid of revolution, the volume of a segment of a hyperboloid of revolution, and the area of a spiral. Heath, Thomas L. 1897. ''Works of Archimedes''. Archimedes' other mathematical achievements include deriving an approximation of pi, defining and in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumcircle
In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polygon has a circumscribed circle. A polygon that does have one is called a cyclic polygon, or sometimes a concyclic polygon because its vertices are concyclic. All triangles, all regular simple polygons, all rectangles, all isosceles trapezoids, and all right kites are cyclic. A related notion is the one of a minimum bounding circle, which is the smallest circle that completely contains the polygon within it, if the circle's center is within the polygon. Every polygon has a unique minimum bounding circle, which may be constructed by a linear time algorithm. Even if a polygon has a circumscribed circle, it may be different from its minimum bounding circle. For example, for an obtuse triangle, the minimum bounding circle has the longest sid ... [...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 flourishing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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