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Isosceles
In geometry, an isosceles triangle () is a triangle that has two sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids. The mathematical study of isosceles triangles dates back to ancient Egyptian mathematics and Babylonian mathematics. Isosceles triangles have been used as decoration from even earlier times, and appear frequently in architecture and design, for instance in the pediments and gables of buildings. The two equal sides are called the ''legs'' and the third side is called the ''base'' of the triangle. The other dimensions of the triangle, such as its height, area, and perimeter, can be calculated b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Trapezoid
In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides (the bases) are parallel, and the two other sides (the legs) are of equal length (properties shared with the parallelogram), and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure (there are in fact two pairs of equal base angles, where one base angle is the supplementary angle of a base angle at the other base). Special cases Trapezoid is defined as a quadrilateral having exactly one pair of parallel sides, with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bipyramid
In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise specified the base Vertex (geometry), vertices are usually coplanar and a bipyramid is usually ''symmetric'', meaning the two pyramids are mirror images across their common base plane. When each apex (geometry), apex (, the off-base vertices) of the bipyramid is on a line perpendicular to the base and passing through its center, it is a ''right'' bipyramid; otherwise it is ''oblique''. When the base is a regular polygon, the bipyramid is also called ''regular''. Definition and properties A bipyramid is a polyhedron constructed by fusing two Pyramid (geometry), pyramids which share the same polygonal base (geometry), base; a pyramid is in turn constructed by connecting each vertex of its base to a single new vertex (geometry), vertex (th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
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 (180 degrees or π radians). 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 the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Set
In discrete geometry, an isosceles set is a set of points with the property that every three of them form an isosceles triangle. More precisely, each three points should determine at most two distances; this also allows degenerate isosceles triangles formed by three equally-spaced points on a line. History The problem of finding the largest isosceles set in a Euclidean space of a given dimension was posed in 1946 by Paul Erdős. In his statement of the problem, Erdős observed that the largest such set in the Euclidean plane has six points. In his 1947 solution, Leroy Milton Kelly showed more strongly that the unique six-point planar isosceles set consists of the vertices and center of a regular pentagon. In three dimensions, Kelly found an eight-point isosceles set, six points of which are the same; the remaining two points lie on a line perpendicular to the pentagon through its center, at the same distance as the pentagon vertices from the center. This three-dimensional example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalene Triangle
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 (180 degrees or π radians). 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 the base and apex is the ''height''. The area of a triangle equals one-half the product of height and base length. In Euclidean geometry, any two points determine a unique line segment situated within a unique straight line, and any three points that do not all lie on the same straight line determine a unique triangle situated within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection Symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In Two-dimensional space, two-dimensional space, there is a line/axis of symmetry, in Three-dimensional space, three-dimensional space, there is a plane (mathematics), plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror image, mirror symmetric. Symmetric function In formal terms, a mathematical object is symmetric with respect to a given mathematical operation, operation such as reflection, Rotational symmetry, rotation, or Translational symmetry, translation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group (algebra), group. Two objects are symmetr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Solid
The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to the vertices of Archimedean solids, and vice versa. One way to construct the Catalan solids is by using the Dorman Luke construction. The Catalan solids are face-transitive or ''isohedral'' meaning that their faces are symmetric to one another, but they are not vertex-transitive because their vertices are not symmetric. Their dual, the Archimedean solids, are vertex-transitive but not face-transitive. Each Catalan solid has constant dihedral angles, meaning the angle between any two adjacent faces is the same. Additionally, two Catalan solids, the rhombic dodecahedron and rhombic triacontahedron, are edge-transitive, meaning their edges are symmetric to each other. Some Catalan solids were discovered by Johannes Kepler during his study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Right Triangle
A special right triangle is a right triangle with some regular feature that makes calculations on the triangle easier, or for which simple formulas exist. For example, a right triangle may have angles that form simple relationships, such as 45°–45°–90°. This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods. Angle-based ''Angle-based'' special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or radians, is equal to the sum of the other two angles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. Properties An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely dominated the field until the early 19th century. His system, now referred to as Euclidean geometry, involved innovations in combination with a synthesis of theories from earlier Greek mathematicians, including Eudoxus of Cnidus, Hippocrates of Chios, Thales and Theaetetus. With Archimedes and Apollonius of Perga, Euclid is generally considered among the greatest mathematicians of antiquity, and one of the most influential in the history of mathematics. Very little is known of Euclid's life, and most information comes from the scholars Proclus and Pappus of Alexandria many centuries later. Medieval Islamic mathematicians invented a fanciful biography, and medieval Byzantine and early Renaissance scholars mistook him for the earlier philo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Obtuse Triangle
An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°). Properties In all triangles, the centroid—the intersection of the medians, each of which connects a vertex with the midpoint of the opposite side—and the incenter—the center of the circle that is internally tangent to all three sides—are in the interior of the triangle. However, while the orthocenter and the circumcenter are in an acute triangle's interior, they are exterior to an obtuse triangle. The orthocenter is the intersection po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
