|
Newton's Theorem (quadrilateral)
In Euclidean geometry Newton's theorem states that in every tangential quadrilateral other than a rhombus, the center of the incircle lies on the Newton line. Statement Let ''ABCD'' be a tangential quadrilateral with at most one pair of parallel sides. Furthermore, let ''E'' and ''F'' the midpoints of its diagonals ''AC'' and ''BD'' and ''P'' be the center of its incircle. Given such a configuration the point P is located on the Newton line, that is line ''EF'' connecting the midpoints of the diagonals. A tangential quadrilateral with two pairs of parallel sides is a rhombus. In this case, both midpoints and the center of the incircle coincide, and by definition, no Newton line exists. Proof Newton's theorem can easily be derived from Anne's theorem considering that in tangential quadrilaterals the combined lengths of opposite sides are equal (Pitot theorem The Pitot theorem in geometry states that in a tangential quadrilateral the two pairs of opposite sides have the sam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Theorem
Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: People * Newton (surname), including a list of people with the surname * Newton (given name), including a list of people with the given name Arts and entertainment * ''Newton'' (film), a 2017 Indian film * Newton (band), Spanish electronic music group * ''Newton'' (Blake), a print by William Blake * ''Newton'' (Paolozzi), a 1995 bronze sculpture by Eduardo Paolozzi * Cecil Newton (''Coronation Street''), a character in the British soap opera ''Coronation Street'' * Curtis Newton, "real" name of pulp magazine character Captain Future * George Newton, a character in the film series ''Beethoven'' * Newton Gearloose, a Disney character, nephew of Gyro Gearloose * Newton, a character in ''The Mighty Hercules'' animated series Places Australia * Newton, South Australia Canada * Newton, Edmonton, Alberta * New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. One of those is the parallel postulate which relates to parallel lines on a Euclidean plane. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logic, logical system in which each result is ''mathematical proof, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangential Quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the Incircle and excircles of a triangle, incircle of the quadrilateral or its inscribed circle, its center is the ''incenter'' and its radius is called the ''inradius''. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called ''circumscribable quadrilaterals'', ''circumscribing quadrilaterals'', and ''circumscriptible quadrilaterals''. Tangential quadrilaterals are a special case of tangential polygons. Other less frequently used names for this class of quadrilaterals are ''inscriptable quadrilateral'', ''inscriptible quadrilateral'', ''inscribable quadrilateral'', ''circumcyclic quadrilateral'', and ''co-cyclic quadrilateral''.. Due to the risk of confusion with a qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incircle
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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Line
In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides. Properties The line segments and that connect the midpoints of opposite sides (the bimedians) of a convex quadrilateral intersect in a point that lies on the Newton line. This point bisects the line segment that connects the diagonal midpoints. By Anne's theorem and its converse, any interior point ''P'' on the Newton line of a quadrilateral has the property that : triangle ABP+ triangle CDP= triangle ADP+ triangle BCP where denotes the area of triangle . If the quadrilateral is a tangential quadrilateral, then its incenter also lies on this line. See also *Complete quadrangle In mathematics, specifically in incidence geometry and especially in projective geometry, a complete quadrangle is a system of geometric objects consisting of any four points in a plane, no three of which are on a common line, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anne's Theorem
In Euclidean geometry, Anne's theorem describes an equality of certain areas within a convex quadrilateral. This theorem is named after the French mathematician Pierre-Léon Anne (1806–1850). Statement The theorem is stated as follows: Let be a convex quadrilateral with diagonals and , that is not a parallelogram. Furthermore, let and be the midpoints of the diagonals, and let be an arbitrary point in the interior of , resulting in that forms four triangles with the edges of . If the two sums of areas of opposite triangles are equal: \left, \triangle BCL \ + \left, \triangle DAL \ = \left, \triangle LAB \ + \left, \triangle DLC \, then the point is located on the Newton line In Euclidean geometry the Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral with at most two parallel sides. Properties The line segments and that connect the midpoints of opposite sides (the ..., that is the line which connects and . F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pitot Theorem
The Pitot theorem in geometry states that in a tangential quadrilateral the two pairs of opposite sides have the same total length. It is named after French engineer Henri Pitot. Statement and converse A tangential quadrilateral is usually defined as a convex polygon, convex quadrilateral for which all four sides are Tangent lines to circles, tangent to the same inscribed circle. Pitot's theorem states that, for these quadrilaterals, the two sums of lengths of opposite sides are the same. Both sums of lengths equal the semiperimeter of the quadrilateral. The converse implication is also true: whenever a convex quadrilateral has pairs of opposite sides with the same sums of lengths, it has an inscribed circle. Therefore, this is an exact characterization: the tangential quadrilaterals are exactly the quadrilaterals with equal sums of opposite side lengths.. See in particular pp. 65–66. Proof idea One way to prove the Pitot's theorem is to divide the sides of any given tang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics 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 11 Dupont 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. Meetings The MAA sponsors the annual summer MathFest and cosponsors with the American Mathematical Society the Joint Mathematics Meeting, held in early January of each year. On occasion the Society for Industrial and Applied Mathematics joins in these meetings. Twenty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
