"Median (geometry) In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposing side, bisecting it. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid. In the case of isosceles and equilateral triangles, a median bisects any angle at a vertex whose two adjacent sides are equal in length. The concept of a median extends to tetrahedra. Each median of a triangle passes through the triangle's centroid, which is the center of mass of an infinitely thin object of uniform"
"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"
"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"
"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,"
"similarity of the two triangles. Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg-Pedoe inequality), named after Daniel Pedoe (1910-1998) and Joseph Jean Baptiste Neuberg (1840-1926), states that if ""a"", ""b"", and ""c"" are the lengths of the sides of a triangle with area ""ƒ"", and ""A"", ""B"", and ""C"" are the lengths of the sides of a triangle with area ""F"", then with equality if and only if the two triangles are similar with pairs of corresponding sides (""A, a""), (""B, b""), and (""C, c""). The expression on the left is not only symmetric under any of the six permutations"
"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"
"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,"
"it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees. Two triangles are said to be ""similar"" if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic theorems about similar triangles are: Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles"
"angle A"" and ""opposed to angle B"". If the lengths of all three sides of a right triangle are integers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple. As with any triangle, the area is equal to one half the base multiplied by the corresponding height. In a right triangle, if one leg is taken as the base then the other is height, so the area of a right triangle is one half the product of the two legs. As a formula the area ""T"" is where ""a"""
"the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another."" Its name may be attributed to its frequent role as the first real test in the ""Elements"" of the intelligence of the reader and as a bridge to the harder propositions that followed. It might also be so named because of the geometrical figure's resemblance to a steep bridge that only a sure-footed donkey could cross. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them"
"Pedoe's inequality In geometry, Pedoe's inequality (also Neuberg-Pedoe inequality), named after Daniel Pedoe (1910-1998) and Joseph Jean Baptiste Neuberg (1840-1926), states that if ""a"", ""b"", and ""c"" are the lengths of the sides of a triangle with area ""ƒ"", and ""A"", ""B"", and ""C"" are the lengths of the sides of a triangle with area ""F"", then with equality if and only if the two triangles are similar with pairs of corresponding sides (""A, a""), (""B, b""), and (""C, c""). The expression on the left is not only symmetric under any of the six permutations of the set { (""A"","
"Heptagonal triangle A heptagonal triangle is an obtuse scalene triangle whose vertices coincide with the first, second, and fourth vertices of a regular heptagon (from an arbitrary starting vertex). Thus its sides coincide with one side and the adjacent shorter and longer diagonals of the regular heptagon. All heptagonal triangles are similar (have the same shape), and so they are collectively known as ""the"" heptagonal triangle. Its angles have measures formula_1 and formula_2 and it is the only triangle with angles in the ratios 1:2:4. The heptagonal triangle has various remarkable properties. The heptagonal triangle's nine-point center is also its"
"Trigonometry Trigonometry (from Greek ""trigōnon"", ""triangle"" and ""metron"", ""measure"") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The 3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be"
"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,"
"was also known to be just one example of a general rule that any triangle where the length of two sides, each squared and then added together , equals the square of the length of the third side , would also be a right angle triangle. This is now known as the Pythagorean theorem, and a triple of numbers that meets this condition is called a Pythagorean triple – both are named after the ancient Greek Pythagoras. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples, and methods for generating such triples have been"
"the third angle. The triangle where two vertices are ideal points and the remaining angle is right, one of the first hyperbolic triangles (1818) described by Ferdinand Karl Schweikart. The triangle where all vertices are ideal points, an ideal triangle is the largest possible triangle in hyperbolic geometry because of the zero sum of the angles. The relations among the angles and sides are analogous to those of spherical trigonometry; the length scale for both spherical geometry and hyperbolic geometry can for example be defined as the length of a side of an equilateral triangle with fixed angles. The length"
"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"
"this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Triangles can be classified according to the lengths of their sides: Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of ""ticks"", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3"
"Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer to a non-square rectangle. A rectangle with vertices ""ABCD"" would be denoted as . The word rectangle comes from the Latin ""rectangulus"", which is a combination of ""rectus"" (as an adjective, right, proper)"
"Sum of angles of a triangle In several geometries, a triangle has three ""vertices"" and three ""sides"", where three angles of a triangle are formed at each vertex by a pair of adjacent sides. In a Euclidean space, the sum of measures of these three angles of any triangle is invariably equal to the straight angle, also expressed as 180 °, radians, two right angles, or a half-turn. It was unknown for a long time whether other geometries exist, where this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the"
"Cathetus In a right triangle, a cathetus (originally from the Greek word ; plural: catheti), commonly known as a leg, is either of the sides that are adjacent to the right angle. It is occasionally called a ""side about the right angle"". The side opposite the right angle is the hypotenuse. In the context of the hypotenuse, the catheti are sometimes referred to simply as ""the other two sides"". If the catheti of a right triangle have equal lengths, the triangle is isosceles. If they have different lengths, a distinction can be made between the minor (shorter) and major (longer)"
"be made quickly and without technical instruments. The geometric law behind the measurement is the Pythagorean theorem (""The square of the hypotenuse of a right triangle is equal to the sum of the squares of the two adjacent sides""). Thales' theorem states that an angle inscribed in a semicircle (with a vertex on the semicircle and its defining rays going through the endpoints of the semicircle) is a right angle. Two application examples in which the right angle and the Thales' theorem are included (see animations). Right angle In geometry and trigonometry, a right angle is an angle of exactly"
"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"
"with sides ""a"", ""b"" and ""c"", and every such triangle has a right angle between the sides of lengths ""a"" and ""b"". An alternative statement is: For any triangle with sides ""a"", ""b"", ""c"", if then the angle between ""a"" and ""b"" measures 90°. This converse also appears in Euclid's ""Elements"" (Book I, Proposition 48): It can be proven using the law of cosines or as follows: Let ""ABC"" be a triangle with side lengths ""a"", ""b"", and ""c"", with Construct a second triangle with sides of length ""a"" and ""b"" containing a right angle. By the Pythagorean theorem, it"
"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"
"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"
"Kepler triangle A Kepler triangle is a right triangle with edge lengths in a geometric progression in which the common ratio is , where is the golden ratio, and can be written: formula_1, or approximately 1 : 1.272 : 1.618. The squares of the edges of this triangle are in geometric progression according to the golden ratio. Triangles with such ratios are named after the German mathematician and astronomer Johannes Kepler (1571–1630), who first demonstrated that this triangle is characterised by a ratio between its short side and hypotenuse equal to the golden ratio. Kepler triangles combine two key mathematical"
"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"
"right triangles. The tilings of rectangles by other tiles which have attracted the most attention are those by congruent non-rectangular polyominoes, allowing all rotations and reflections. There are also tilings by congruent polyaboloes. Rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term oblong is occasionally used to refer"
"Pythagorean theorem In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides ""a"", ""b"" and ""c"", often called the ""Pythagorean equation"": where ""c"" represents the length of the hypotenuse and ""a"" and ""b"" the lengths of the triangle's other two sides. Although it is"
"congruent angles. AA postulate In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180-(32+64)=84. (This is sometimes referred to as the AAA Postulate—which is true in all respects, but two angles are entirely sufficient.) The postulate can be better understood by working in reverse order. The two triangles on"
"AA postulate In Euclidean geometry, the AA postulate states that two triangles are similar if they have two corresponding angles congruent. The AA postulate follows from the fact that the sum of the interior angles of a triangle is always equal to 180°. By knowing two angles, such as 32° and 64° degrees, we know that the next angle is 84°, because 180-(32+64)=84. (This is sometimes referred to as the AAA Postulate—which is true in all respects, but two angles are entirely sufficient.) The postulate can be better understood by working in reverse order. The two triangles on grids A"