"only on the angle between its two legs. Euclid defined an isosceles triangle as a triangle with exactly two equal sides, but modern treatments prefer to define isosceles triangles as having at least two equal sides. The difference between these two definitions is that the modern version makes equilateral triangles (with three equal sides) a special case of isosceles triangles. A triangle that is not isosceles (having three unequal sides) is called scalene. ""Isosceles"" is a compound word, made from the Greek roots ""isos"" (equal) and ""skelos"" (leg). The same word is used, for instance, for isosceles trapezoids, trapezoids with"
"end at an ideal vertex represented as an ""omega point"". Such a pair of sides may also be said to form an angle of zero. A triangle with a zero angle is impossible in Euclidean geometry for straight sides lying on distinct lines. However, such zero angles are possible with tangent circles. A triangle with one ideal vertex is called an omega triangle. Special Triangles with ideal vertices are: A triangle where one vertex is an ideal point, one angle is right: the third angle is the angle of parallelism for the length of the side between the right and"
"two equal sides, and for isosceles sets, sets of points every three of which form an isosceles triangle. In an isosceles triangle that has exactly two equal sides, the equal sides are called legs and the third side is called the base. The angle included by the legs is called the ""vertex angle"" and the angles that have the base as one of their sides are called the ""base angles"". The vertex opposite the base is called the apex. In the equilateral triangle case, since all sides are equal, any side can be called the base. Whether an isosceles triangle"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"Right triangle A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the ""hypotenuse"" (side ""c"" in the figure). The sides adjacent to the right angle are called ""legs"" (or ""catheti"", singular: ""cathetus""). Side ""a"" may be identified as the side ""adjacent to angle B"" and ""opposed to"" (or ""opposite"") ""angle A"", while side ""b"" is the side ""adjacent to"
"A triangle (""X"", ""Y"", ""Z"", ""u"", ""v"", ""w"") consists of 3 objects ""X"", ""Y"", and ""Z"", together with morphisms ""u"" : ""X"" → ""Y"", ""v"" : ""Y"" → ""Z"" and ""w"" : ""Z"" → ""X""[1]. Triangles are generally written in the unravelled form: Triangulated category In mathematics, a triangulated category is a category together with the additional structure of a ""translation functor"" and a class of ""distinguished triangles"". Prominent examples are the derived category of an abelian category and the stable homotopy category of spectra (more generally, the homotopy category of a stable ∞-category), both of which carry the structure"
"divides this segment in the ratio (when taken from the short to the long side) If the angle bisectors to angles ""A"" and ""B"" intersect at ""P"", and the angle bisectors to angles ""C"" and ""D"" intersect at ""Q"", then The term ""trapezoid"" was once defined as a quadrilateral without any parallel sides in Britain and elsewhere. (The Oxford English Dictionary says ""Often called by English writers in the 19th century"".) According to the ""Oxford English Dictionary"", the sense of a figure with no sides parallel is the meaning for which Proclus introduced the term ""trapezoid"". This is retained in"
"product of the two shorter sides of the triangle, at least one of which must be even. An example of a Heronian triangle which is not right-angled is the isosceles triangle with sidelengths 5, 5, and 6, whose area is 12. This triangle is obtained by joining two copies of the right-angled triangle with sides 3, 4, and 5 along the sides of length 4. This approach works in general, as illustrated in the adjacent picture. One takes a Pythagorean triple (""a"", ""b"", ""c""), with ""c"" being largest, then another one (""a"", ""d"", ""e""), with ""e"" being largest, constructs the"
"Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon, so it is also referred to as a regular triangle. Denoting the common length of the sides of the equilateral triangle as ""a"", we can determine using the Pythagorean theorem that: Denoting the radius of the circumscribed circle as ""R"", we can determine using trigonometry that: Many of these quantities"
"Triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ""A"", ""B"", and ""C"" is denoted formula_1. In Euclidean geometry any three points, when non-collinear, determine a unique triangle and simultaneously, a unique 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,"
"diagonals , although there exist non-manifold surfaces with no diagonals and any number of vertices greater than five . used Schönhardt's polyhedron as the basis for a proof that it is NP-complete to determine whether a non-convex polyhedron can be triangulated. Schönhardt polyhedron In geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices. It is named after German mathematician Erich Schönhardt, who first described it in 1928. The Schönhardt polyhedron can be formed by two congruent equilateral triangles in two parallel planes, such that the line through the centers of"
"triangles would be a completely flat figure, and a heptagonal or higher would have the triangles not meet at all. Right pyramids with regular star polygon bases are called star pyramids. For example, the pentagrammic pyramid has a pentagram base and 5 intersecting triangle sides. A right pyramid can be named as ( )∨P, where ( ) is the apex point, ∨ is a join operator, and P is a base polygon. An isosceles triangle right tetrahedron can be written as ( )∨[( )∨{ }] as the join of a point to an isosceles triangle base, as [( )∨( )]∨{"
"Pons asinorum In geometry, the statement that the angles opposite the equal sides of an isosceles triangle are themselves equal is known as the pons asinorum (; ), typically translated as ""bridge of asses"". This statement is Proposition 5 of Book 1 in Euclid's ""Elements"", and is also known as the isosceles triangle theorem. Its converse is also true: if two angles of a triangle are equal, then the sides opposite them are also equal. The name of this statement is also used metaphorically for a problem or challenge which will separate the sure of mind from the simple, the"
"a triangle with any two sides ""a"" and ""b"", and an angle ""C"" between them is Each angle of an equilateral triangle is 60°, so The sine of 60° is formula_50. Thus since all sides of an equilateral triangle are equal. Equilateral triangles have frequently appeared in man made constructions: Equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides are equal. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°. It is also a regular polygon,"
"Isosceles triangle In geometry, an isosceles triangle is a triangle that has two sides of equal length. 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"
"Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two Latin words ""quadri"", a variant of four, and ""latus"", meaning ""side"". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ""ABCD"" add up to 360 degrees of arc, that is"
"of recognition of the concept of ""betweenness"" and the resulting ambiguity of ""inside"" versus ""outside"" of figures. Isosceles triangle In geometry, an isosceles triangle is a triangle that has two sides of equal length. 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"
"axioms. Around 1830 though, the Hungarian János Bolyai and the Russian Nikolai Ivanovich Lobachevsky separately published treatises on a type of geometry that does not include the parallel postulate, called hyperbolic geometry. In this geometry, an infinite number of parallel lines pass through the point ""P"". Consequently, the sum of angles in a triangle is less than 180° and the ratio of a circle's circumference to its diameter is greater than pi. In the 1850s, Bernhard Riemann developed an equivalent theory of elliptical geometry, in which no parallel lines pass through ""P"". In this geometry, triangles have more than 180°"
"Trapezoid In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid () in American and Canadian English but as a trapezium () in English outside North America. The parallel sides are called the ""bases"" of the trapezoid and the other two sides are called the ""legs"" or the lateral sides (if they are not parallel; otherwise there are two pairs of bases). A ""scalene trapezoid"" is a trapezoid with no sides of equal measure, in contrast to the special cases below. The term ""trapezium"" has been in use in English"
"by moving the triangle, rotating it, or reflecting it as in a mirror, any of which give a congruent triangle, or even by rescaling it to give a similar triangle: A non-planar triangle is a triangle which is not contained in a (flat) plane. Some examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry. While the measures of the internal angles in planar triangles always sum to 180°, a hyperbolic triangle has measures of angles that sum to less than 180°, and a spherical triangle has measures of angles that"
"Nonagon In geometry, a nonagon () or enneagon () is a nine-sided polygon or 9-gon. The name ""nonagon"" is a prefix hybrid formation, from Latin (""nonus"", ""ninth"" + ""gonon""), used equivalently, attested already in the 16th century in French ""nonogone"" and in English from the 17th century. The name ""enneagon"" comes from Greek ""enneagonon"" (εννεα, ""nine"" + γωνον (from γωνία = ""corner"")), and is arguably more correct, though less common than ""nonagon"". A ""regular nonagon"" is represented by Schläfli symbol {9} and has internal angles of 140°. The area of a regular nonagon of side length ""a"" is given by"
"their internal angles, measured here in degrees. A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and such a triangle is therefore equilateral. Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a ""2-simplex"" (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4"
"whose bases do not lie in parallel planes would be called a truncated cylinder. From a polyhedral viewpoint, a cylinder can also be seen as a dual of a bicone as an infinite-sided bipyramid. Cylinder A cylinder (from Greek κύλινδρος – ""kulindros"", ""roller, tumbler"") has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on top and bottom. This traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and"
"equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26). Triangles with three equal angles (AAA) are similar, but not necessarily congruent. Also, triangles with two equal sides and an adjacent angle are not necessarily equal or congruent. The sum of the angles of a triangle is equal to a straight angle (180 degrees). This causes an equilateral triangle to have three interior angles of 60 degrees. Also, it causes every triangle to have at least two acute angles and up to one obtuse or right angle. The celebrated Pythagorean theorem (book I,"
"have an edge are typically excluded from consideration. However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent (""bounded"") and complete. The term ""without boundary"" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are"
"inequality. An important characteristic of the automedian triangle is that the squares of its sides form an arithmetic progression. Specifically, formula_91 so formula_92. A triangle family with integer sides formula_93 and with rational bisector formula_94 of angle A is given by with integers formula_99. There exist infinitely many non-similar triangles in which the three sides and the bisectors of each of the three angles are integers. There exist infinitely many non-similar triangles in which the three sides and the two trisectors of each of the three angles are integers. However, for ""n"" > 3 there exist no triangles in which"
"triangle or the Calabi triangle. Thus, the Calabi triangle may be defined as a triangle that is not equilateral and has three placements for its largest square. The Calabi triangle is isosceles. The ratio of the base to either leg is which approximates to 1.55138752454. In terms of trigonometric functions, it is It is the largest positive root of and has continued fraction representation [1, 1, 1, 4, 2, 1, 2, 1, 5, 2, 1, 3, 1, 1, 390, ...]. The Calabi triangle is obtuse with base angles 39.1320261...° and third angle 101.7359477...°. Calabi triangle The Calabi triangle is a"
"is a power of multiplied by the product of one or more distinct Fermat primes, none of which divides . In the case of trisection (, which is a Fermat prime), this condition becomes the above-mentioned requirement that not be divisible by . Angle trisection Angle trisection is a classical problem of compass and straightedge constructions of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. The problem as stated is impossible to solve for arbitrary angles, as proved by Pierre"
"triangle problem is the equilateral triangle. In general, the polygon with sides having the largest area and a given perimeter is the regular polygon, which is closer to being a circle than is any irregular polygon with the same number of sides. The word comes from the Greek περίμετρος ""perimetros"" from περί ""peri"" ""around"" and μέτρον ""metron"" ""measure"". Perimeter A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path or its length—it can be thought of as the length of the outline of a shape. The perimeter of a circle or"
"Heronian triangle In geometry, a Heronian triangle is a triangle that has side lengths and area that are all integers. Heronian triangles are named after Hero of Alexandria. The term is sometimes applied more widely to triangles whose sides and area are all rational numbers, since one can rescale the sides by a common multiple to obtain a triangle that is Heronian in the above sense. Any right-angled triangle whose sidelengths are a Pythagorean triple is a Heronian triangle, as the side lengths of such a triangle are integers, and its area is also an integer, being half of the"
"angle is ""d""/""n"" of the half-circle. ""2"" means a right triangle. When these are whole numbers, the triangle is called a Möbius triangle, and corresponds to a ""non""-overlapping tiling, and the symmetry group is called a triangle group. In the sphere there are three Möbius triangles plus one one-parameter family; in the plane there are three Möbius triangles, while in hyperbolic space there is a three-parameter family of Möbius triangles, and no exceptional objects. A fundamental domain triangle (""p"" ""q"" ""r""), with vertex angles /""p"", /""q"", and /""r"", can exist in different spaces depending on the value of the sum"
"or hypercycle, then the triangle has no circumscribed circle. As in spherical and elliptical geometry, in hyperbolic geometry if two triangles are similar, they must be congruent. A special polygon in hyperbolic geometry is the regular apeirogon, a uniform polygon with an infinite number of sides. In Euclidean geometry, the only way to construct such a polygon is to make the side lengths tend to zero and the apeirogon is indistinguishable from a circle, or make the interior angles tend to 180 degrees and the apeirogon approaches a straight line. However, in hyperbolic geometry, a regular apeirogon has sides of"
"Golden triangle (mathematics) A golden triangle, also known as the sublime triangle,is an isosceles triangle in which the duplicated side is in the golden ratio to the distinct side: Golden triangles are found in the nets of several stellations of dodecahedrons and icosahedrons. Also, it is the shape of the triangles found in the points of pentagrams. The vertex angle is equal to Since the angles of a triangle sum to 180°, base angles are therefore 72° each. The golden triangle can also be found in a regular decagon, or an equiangular and equilateral ten-sided polygon, by connecting any two"