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A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle. The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between base and apex is the height. The area of a triangle equals one-half the product of height and base length.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
In geometry, bisection is the division of something into two equal or congruent parts. Usually it involves a bisecting line, also 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 . In three-dimensional space, bisection is usually done by a bisecting plane, also called the bisector.
In geometry, an altitude of a triangle is a line segment through a given vertex and perpendicular to a line containing the side or edge opposite the apex. This (infinite) line containing the (finite) base is called the extended base of the altitude. The intersection of the extended base and the altitude is called the foot of the altitude. The length of the altitude, often simply called "the altitude", is the distance between the foot and the apex. The process of drawing the altitude from a vertex to the foot is known as dropping the altitude at that vertex. It is a special case of orthogonal projection.
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In geometry, an orthocentric system is a set of four points on a plane, one of which is the orthocenter of the triangle formed by the other three. Equivalently, the lines passing through disjoint pairs among the points are perpendicular, and the four circles passing through any three of the four points have the same radius.
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Fagnano may refer to
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In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775:
For a given acute triangle determine the inscribed triangle of minimal perimeter.
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
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Toschi may refer to:
In geometry, the tangential triangle of a reference triangle is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at the reference triangle's vertices. Thus the incircle of the tangential triangle coincides with the circumcircle of the reference triangle.
In Euclidean geometry, the Clawson point is a special point in a triangle defined by the trilinear coordinates tan α : tan β : tan γ, where α, β, γ are the interior angles at the triangle vertices A, B, C. It is named after John Wentworth Clawson, who published it 1925 in the American Mathematical Monthly. It is denoted X(19) in Clark Kimberling's Encyclopedia of Triangle Centers.
References
- 1 2 3 4 O'Connor, John J.; Robertson, Edmund F., "Giovanni Francesco Fagnano dei Toschi", MacTutor History of Mathematics Archive , University of St Andrews
- ↑
This article incorporates text from a publication now in the public domain : Herbermann, Charles, ed. (1913). "Giulio Carlo de' Toschi di Fagnano". Catholic Encyclopedia . New York: Robert Appleton Company. - ↑ Gutkin, Eugene (1997), "Two applications of calculus to triangular billiards", The American Mathematical Monthly, 104 (7): 618–622, doi:10.2307/2975055, JSTOR 2975055, MR 1468291 .
- ↑ Cieslik, Dietmar (2006), Shortest Connectivity: An Introduction with Applications in Phylogeny, Combinatorial Optimization, vol. 17, Springer, p. 6, ISBN 9780387235394
- ↑ Plastria, Frank (2006), "Four-point Fermat location problems revisited. New proofs and extensions of old results" (PDF), IMA Journal of Management Mathematics, 17 (4): 387–396, doi:10.1093/imaman/dpl007, Zbl 1126.90046, archived from the original (PDF) on 2016-03-04, retrieved 2014-05-18
- ↑ Fagnano, G. F. (1775), "Problemata quaedam ad methodum maximorum et minimorum spectantia", Nova Acta Eruditorum: 281–303.
- ↑ http://www.izwtalt.uni-wuppertal.de/Acta/NAE1775.pdf [ bare URL PDF ]
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