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Thales' Theorem
In geometry, Thales's theorem states that if , , and are distinct points on a circle where the line is a diameter, the angle is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's ''Euclid's Elements, Elements''. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras. History Babylonian mathematics, Babylonian mathematicians knew this for special cases before Greek mathematicians proved it. Thales of Miletus (early 6th century BC) is traditionally credited with proving the theorem; however, even by the 5th century BC there was nothing extant of Thales' writing, and inventions and ideas were attributed to men of wisdom such as Thales and Pythagoras by later doxography, doxographers based on hearsay and speculation. Reference to Thales was made by Proclus (5th century AD), and by Diogenes Laërtius (3rd century AD) docum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diogenes Laërtius
Diogenes Laërtius ( ; , ; ) was a biographer of the Greek philosophers. Little is definitively known about his life, but his surviving book ''Lives and Opinions of Eminent Philosophers'' is a principal source for the history of ancient Greek philosophy. His reputation is controversial among scholars because he often repeats information from his sources without critically evaluating it. In many cases, he focuses on insignificant details of his subjects' lives while ignoring important details of their philosophical teachings and he sometimes fails to distinguish between earlier and later teachings of specific philosophical schools. However, unlike many other ancient secondary sources, Diogenes Laërtius tends to report philosophical teachings without trying to reinterpret or expand on them, and so his accounts are often closer to the primary sources. Due to the loss of so many of the primary sources on which Diogenes relied, his work has become the foremost surviving source on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thales Theorem By Refelection1
Thales of Miletus ( ; ; ) was an Ancient Greek pre-Socratic philosopher from Miletus in Ionia, Asia Minor. Thales was one of the Seven Sages, founding figures of Ancient Greece. Beginning in eighteenth-century historiography, many came to regard him as the first philosopher in the Greek tradition, breaking from the prior use of mythology to explain the world and instead using natural philosophy. He is thus otherwise referred to as the first to have engaged in mathematics, science, and deductive reasoning. Thales's view that all of nature is based on the existence of a single ultimate substance, which he theorized to be water, was widely influential among the philosophers of his time. Thales thought the Earth floated on water. In mathematics, Thales is the namesake of Thales's theorem, and the intercept theorem can also be referred to as Thales's theorem. Thales was said to have calculated the heights of the pyramids and the distance of ships from the shore. In scienc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pythagorean Trigonometric Identity
The Pythagorean trigonometric identity, also called simply the Pythagorean identity, is an identity expressing the Pythagorean theorem in terms of trigonometric functions. Along with the sum-of-angles formulae, it is one of the basic relations between the sine and cosine functions. The identity is :\sin^2 \theta + \cos^2 \theta = 1. As usual, \sin^2 \theta means (\sin\theta)^2. Proofs and their relationships to the Pythagorean theorem Proof based on right-angle triangles Any similar triangles have the property that if we select the same angle in all of them, the ratio of the two sides defining the angle is the same regardless of which similar triangle is selected, regardless of its actual size: the ratios depend upon the three angles, not the lengths of the sides. Thus for either of the similar right triangles in the figure, the ratio of its horizontal side to its hypotenuse is the same, namely . The elementary definitions of the sine and cosine functions in terms of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a Line (mathematics), line is a number that describes the direction (geometry), direction of the line on a plane (geometry), plane. Often denoted by the letter ''m'', slope is calculated as the ratio of the vertical change to the horizontal change ("rise over run") between two distinct points on the line, giving the same number for any choice of points. The line may be physical – as set by a Surveying, road surveyor, pictorial as in a diagram of a road or roof, or Pure mathematics, abstract. An application of the mathematical concept is found in the grade (slope), grade or gradient in geography and civil engineering. The ''steepness'', incline, or grade of a line is the absolute value of its slope: greater absolute value indicates a steeper line. The line trend is defined as follows: *An "increasing" or "ascending" line goes from left to right and has positive slope: m>0. *A "decreasing" or "descending" line goes from left to right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosceles Triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), 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 (mathematics), 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 (geometry), ''base'' of the triangle. The other dimensions of the triangle, such ... [...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] |
Sum Of Angles Of A Triangle
In a Euclidean space, the sum of angles of a triangle equals a straight angle (180 degrees, radians, two right angles, or a half- turn). A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides. The sum can be computed directly using the definition of angle based on the dot product and trigonometric identities, or more quickly by reducing to the two-dimensional case and using Euler's identity. It was unknown for a long time whether other geometries exist, for which this sum is different. The influence of this problem on mathematics was particularly strong during the 19th century. Ultimately, the answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle. Its difference from 180° is a case of ''angular defect'' and serves as an important distinction for geometric systems. Cases Euclidean geometry In Euclidean geometry, the triangle postulate states that the sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paradiso (Dante)
''Paradiso'' (; Italian for "Paradise" or "Heaven (Christianity), Heaven") is the third and final part of Dante's ''Divine Comedy'', following the ''Inferno (Dante), Inferno'' and the ''Purgatorio''. It is an allegory telling of Dante's journey through Heaven, guided by Beatrice Portinari, Beatrice, who symbolises theology. In the poem, Paradise is depicted as a series of concentric spheres surrounding the Earth, consisting of the Moon, Mercury (planet), Mercury, Venus, the Sun, Mars, Jupiter, Saturn, the Fixed stars, Fixed Stars, the Primum Mobile and finally, the Empyrean. It was written in the early 14th century. Allegorically, the poem represents the soul's ascent to God. Introduction The ''Paradiso'' begins at the top of Purgatorio, Mount Purgatory, called the Purgatorio#The Earthly Paradise, Earthly Paradise (i.e. the Garden of Eden), at noon on Wednesday, March 30 (or April 13), 1300, following Easter Sunday. Dante's journey through Paradise takes approximately twenty-f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dante Alighieri
Dante Alighieri (; most likely baptized Durante di Alighiero degli Alighieri; – September 14, 1321), widely known mononymously as Dante, was an Italian Italian poetry, poet, writer, and philosopher. His ''Divine Comedy'', originally called (modern Italian: ) and later christened by Giovanni Boccaccio, is widely considered one of the most important poems of the Middle Ages and the greatest literary work in the Italian language. Dante chose to write in the vernacular, specifically, his own Tuscan dialect, at a time when much literature was still written in Latin, which was accessible only to educated readers, and many of his fellow Italian poets wrote in French or Provençal dialect, Provençal. His ' (''On Eloquence in the Vernacular'') was one of the first scholarly defenses of the vernacular. His use of the Florentine dialect for works such as ''La Vita Nuova, The New Life'' (1295) and ''Divine Comedy'' helped establish the modern-day standardized Italian language. His wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
