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Butterfly Theorem
The butterfly theorem is a classical result in Euclidean geometry, which can be stated as follows:Johnson, Roger A., ''Advanced Euclidean Geometry'', Dover Publ., 2007 (orig. 1929). Let be the midpoint of a chord of a circle, through which two other chords and are drawn; and intersect chord at and correspondingly. Then is the midpoint of . Proof A formal proof of the theorem is as follows: Let the perpendiculars and be dropped from the point on the straight lines and respectively. Similarly, let and be dropped from the point perpendicular to the straight lines and respectively. Since :: \triangle MXX' \sim \triangle MYY', : = , :: \triangle MXX'' \sim \triangle MYY'', : = , :: \triangle AXX' \sim \triangle CYY'', : = , :: \triangle DXX'' \sim \triangle BYY', : = . From the preceding equations and the intersecting chords theorem, it can be seen that : \left(\right)^2 = , : = , : = , : = , : = , since . So : = . ... [...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 geom ... [...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] |
Chord (geometry)
A chord of a circle is a straight line segment whose endpoints both lie on a circular arc. The infinite line extension of a chord is a secant line, or just ''secant''. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word ''chord'' is from the Latin ''chorda'' meaning '' bowstring''. In circles Among properties of chords of a circle are the following: # Chords are equidistant from the center if and only if their lengths are equal. # Equal chords are subtended by equal angles from the center of the circle. # A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle. # If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem). In conics The midpoints of a set of parallel chords of a coni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendiculars
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can be defined between two lines (or two line segments), between a line and a plane, and between two planes. Perpendicularity is one particular instance of the more general mathematical concept of ''orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its ''normal vector''. Definitions A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersecting Chords Theorem
The intersecting chords theorem or just the chord theorem is a statement in elementary geometry that describes a relation of the four line segments created by two intersecting chords within a circle. It states that the products of the lengths of the line segments on each chord are equal. It is Proposition 35 of Book 3 of Euclid's ''Elements''. More precisely, for two chords ''AC'' and ''BD'' intersecting in a point ''S'' the following equation holds: :, AS, \cdot, SC, =, BS, \cdot, SD, The converse is true as well, that is if for two line segments ''AC'' and ''BD'' intersecting in S the equation above holds true, then their four endpoints ''A'', ''B'', ''C'' and ''D'' lie on a common circle. Or in other words if the diagonals of a quadrilateral ''ABCD'' intersect in ''S'' and fulfill the equation above then it is a cyclic quadrilateral. The value of the two products in the chord theorem depends only on the distance of the intersection point ''S'' from the circle's center ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forum Geometricorum
''Forum Geometricorum: A Journal on Classical Euclidean Geometry'' is a peer-reviewed open-access academic journal that specializes in mathematical research papers on Euclidean geometry. It was founded in 2001, is published by Florida Atlantic University, and is indexed among others by Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also ... and . Its founding editor-in-chief was Paul Yiu, a professor of mathematics at Florida Atlantic, now retired. All papers are available online immediately upon acceptance through the journal's web site. , Forum Geometricorum is no longer accepting submissions. Prior issues are still available. See also * International Journal of Geometry References External links * {{official website, http://forumgeom.fau.edu/ Mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
William Wallace (mathematician)
William Wallace LLD (23 September 176828 April 1843) was a Scottish mathematician and astronomer who invented the eidograph (an improved pantograph). Life Wallace was born at Dysart in Fife, the son of Alexander Wallace, a leather manufacturer, and his wife, Janet Simson. He received his school education in Dysart and Kirkcaldy. In 1784 his family moved to Edinburgh, where he himself was set to learn the trade of a bookbinder. In 1790 he appears as "William Wallace, bookbinder" living and trading at Cowgatehead, at the east end of the Grassmarket. His taste for mathematics had already developed itself, and he made such use of his leisure hours that before the completion of his apprenticeship he had made considerable acquirements in geometry, algebra and astronomy. He was further assisted in his studies by John Robison (1739–1805) and John Playfair, to whom his abilities had become known. After various changes of situation, dictated mainly by a desire to gain time for st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sir William Herschel
Frederick William Herschel (; german: Friedrich Wilhelm Herschel; 15 November 1738 – 25 August 1822) was a German-born British astronomer and composer. He frequently collaborated with his younger sister and fellow astronomer Caroline Herschel (1750–1848). Born in the Electorate of Hanover, William Herschel followed his father into the military band of Hanover, before emigrating to Great Britain in 1757 at the age of nineteen. Herschel constructed his first large telescope in 1774, after which he spent nine years carrying out sky surveys to investigate double stars. Herschel published catalogues of nebulae in 1802 (2,500 objects) and in 1820 (5,000 objects). The resolving power of the Herschel telescopes revealed that many objects called nebulae in the Messier catalogue were actually clusters of stars. On 13 March 1781 while making observations he made note of a new object in the constellation of Gemini. This would, after several weeks of verification and consultati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996.Interview with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
