Ex-tangential quadrilateral
In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has been called an exscriptible quadrilateral; the circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors; these are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is related to the tangential quadrilateral. Another name for an excircle is an escribed circle, but that name has been used for a circle tangent to one side of a convex quadrilateral and the extensions of the adjacent two sides. In that context all convex quadrilaterals have four escribed circles, but they can at most have one excircle. Kites are examples of ex-tangential quadrilaterals.
Parallelograms can be considered ex-tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides. Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths. A convex quadrilateral is ex-tangential; these are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, the external angle bisectors at the angles formed where the extensions of opposite sides intersect. For the purpose of calculation, a more useful characterization is that a convex quadrilateral with successive sides a, b, c, d is ex-tangential if and only if the sum of two adjacent sides is equal to the sum of the other two sides; this is possible in two different ways—either as a + b = c + d or a + d = b + c. This was proved by Jakob Steiner in 1846.
In the first case, the excircle is outside the biggest of the vertices A or C, whereas in the second case it is outside the biggest of the vertices B or D, provided that the sides of the quadrilateral ABCD are a = AB, b = BC, c = CD, d = DA. A way of combining these characterizations regarding the sides is that the absolute values of the differences between opposite sides are equal for the two pairs of opposite sides, | a − c | = | b − d |; these equations are related to the Pitot theorem for tangential quadrilaterals, where the sums of opposite sides are equal for the two pairs of opposite sides. If opposite sides in a convex quadrilateral ABCD intersect at E and F A B + B C = A D + D C ⇔ A E + E C = A F + F C; the implication to the right is named after L. M. Urquhart although it was proved long before by Augustus De Morgan in 1841. Daniel Pedoe named it the most elementary theorem in Euclidean geometry since it only concerns straight lines and distances; that there in fact is an equivalence was proved by Mowaffac Hajja, which makes the equality to the right another necessary and sufficient condition for a quadrilateral to be ex-tangential.
A few of the metric characterizations of tangential quadrilaterals have similar counterparts for ex-tangential quadrilaterals, as can be seen in the table below. Thus a convex quadrilateral has an incircle or an excircle outside the appropriate vertex if and only if any one of the five necessary and sufficient conditions below is satisfied; the notations in this table are as follows: In a convex quadrilateral ABCD, the diagonals intersect at P. R1, R2, R3, R4 are the circumradii in triangles ABP, BCP, CDP, DAP. An ex-tangential quadrilateral ABCD with sides a, b, c, d has area K = a b c d sin B + D 2. Note that this is the same formula as the one for the area of a tangential quadrilateral and it is derived from Bretschneider's formula in the same way; the exradius for an ex-tangential quadrilateral with consecutive sides a, b, c, d is given by r = K | a − c | = K | b −
Parallel (geometry)
In geometry, parallel lines are lines in a plane which do not meet. By extension, a line and a plane, or two planes, in three-dimensional Euclidean space that do not share a point are said to be parallel. However, two lines in three-dimensional space which do not meet must be in a common plane to be considered parallel. Parallel planes are planes in the same three-dimensional space. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism; the parallel symbol is ∥. For example, A B ∥ C D indicates that line AB is parallel to line CD. In the Unicode character set, the "parallel" and "not parallel" signs have codepoints U+2225 and U+2226, respectively. In addition, U+22D5 represents the relation "equal and parallel to". Given parallel straight lines l and m in Euclidean space, the following properties are equivalent: Every point on line m is located at the same distance from line l.
Line m is in the same plane as line l but does not intersect l. When lines m and l are both intersected by a third straight line in the same plane, the corresponding angles of intersection with the transversal are congruent. Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, so, are "more complicated" than the second. Thus, the second property is the one chosen as the defining property of parallel lines in Euclidean geometry; the other properties are consequences of Euclid's Parallel Postulate. Another property that involves measurement is that lines parallel to each other have the same gradient; the definition of parallel lines as a pair of straight lines in a plane which do not meet appears as Definition 23 in Book I of Euclid's Elements. Alternative definitions were discussed by other Greeks as part of an attempt to prove the parallel postulate. Proclus attributes a definition of parallel lines as equidistant lines to Posidonius and quotes Geminus in a similar vein.
Simplicius mentions Posidonius' definition as well as its modification by the philosopher Aganis. At the end of the nineteenth century, in England, Euclid's Elements was still the standard textbook in secondary schools; the traditional treatment of geometry was being pressured to change by the new developments in projective geometry and non-Euclidean geometry, so several new textbooks for the teaching of geometry were written at this time. A major difference between these reform texts, both between themselves and between them and Euclid, is the treatment of parallel lines; these reform texts were not without their critics and one of them, Charles Dodgson, wrote a play and His Modern Rivals, in which these texts are lambasted. One of the early reform textbooks was James Maurice Wilson's Elementary Geometry of 1868. Wilson based his definition of parallel lines on the primitive notion of direction. According to Wilhelm Killing the idea may be traced back to Leibniz. Wilson, without defining direction since it is a primitive, uses the term in other definitions such as his sixth definition, "Two straight lines that meet one another have different directions, the difference of their directions is the angle between them."
Wilson In definition 15 he introduces parallel lines in this way. Wilson Augustus De Morgan reviewed this text and declared it a failure on the basis of this definition and the way Wilson used it to prove things about parallel lines. Dodgson devotes a large section of his play to denouncing Wilson's treatment of parallels. Wilson edited this concept out of the third and higher editions of his text. Other properties, proposed by other reformers, used as replacements for the definition of parallel lines, did not fare much better; the main difficulty, as pointed out by Dodgson, was that to use them in this way required additional axioms to be added to the system. The equidistant line definition of Posidonius, expounded by Francis Cuthbertson in his 1874 text Euclidean Geometry suffers from the problem that the points that are found at a fixed given distance on one side of a straight line must be shown to form a straight line; this must be assumed to be true. The corresponding angles formed by a transversal property, used by W. D. Cooley in his 1860 text, The Elements of Geometry and explained requires a proof of the fact that if one transversal meets a pair of lines in congruent corresponding angles all transversals must do so.
Again, a new axiom is needed to justify this statement. The three properties above lead to three different methods of construction of parallel lines; because parallel lines in a Euclidean plane are equidistant there is a unique distance between the two parallel lines. Given the equations of two non-vertical, non-horizontal parallel lines, y = m x + b 1 y = m x + b 2
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work to solve mathematical problems. Mathematics is concerned with numbers, quantity, space and change. One of the earliest known mathematicians was Thales of Miletus, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number", it was the Pythagoreans who coined the term "mathematics", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria, she succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells.
Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages was ongoing throughout the reign of certain caliphs, it turned out that certain scholars became experts in the works they translated and in turn received further support for continuing to develop certain sciences; as these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham; the Renaissance brought an increased emphasis on science to Europe.
During this period of transition from a feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli. As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.” In 1810, Humboldt convinced the King of Prussia to build a university in Berlin based on Friedrich Schleiermacher’s liberal ideas. Thus and laboratories started to evolve. British universities of this period adopted some approaches familiar to the Italian and German universities, but as they enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences that inspired Humboldt.
The Universities of Oxford and Cambridge emphasized the importance of research, arguably more authentically implementing Humboldt’s idea of a university than German universities, which were subject to state authority. Overall, science became the focus of universities in the 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge; the German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of “freedom of scientific research and study.” Mathematicians cover a breadth of topics within mathematics in their undergraduate education, proceed to specialize in topics of their own choice at the graduate level.
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics. Mathematicians involved with solving problems with applications in real life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, localized constructs, applied mathematicians work in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM careers; the discipline of applied mathematics concerns
Diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal; the word "diagonal" derives from the ancient Greek διαγώνιος diagonios, "from angle to angle". In matrix algebra, a diagonal of a square matrix is a set of entries extending from one corner to the farthest corner. There are other, non-mathematical uses. In engineering, a diagonal brace is a beam used to brace a rectangular structure to withstand strong forces pushing into it. Diagonal pliers are wire-cutting pliers defined by the cutting edges of the jaws intersects the joint rivet at an angle or "on a diagonal", hence the name. A diagonal lashing is a type of lashing used to bind spars or poles together applied so that the lashings cross over the poles at an angle. In association football, the diagonal system of control is the method referees and assistant referees use to position themselves in one of the four quadrants of the pitch.
As applied to a polygon, a diagonal is a line segment joining any two non-consecutive vertices. Therefore, a quadrilateral has two diagonals, joining opposite pairs of vertices. For any convex polygon, all the diagonals are inside the polygon, but for re-entrant polygons, some diagonals are outside of the polygon. Any n-sided polygon, convex or concave, has n 2 diagonals, as each vertex has diagonals to all other vertices except itself and the two adjacent vertices, or n − 3 diagonals, each diagonal is shared by two vertices. In a convex polygon, if no three diagonals are concurrent at a single point in the interior, the number of regions that the diagonals divide the interior into is given by + = 24. For n-gons with n=3, 4... the number of regions is 1, 4, 11, 25, 50, 91, 154, 246... This is OEIS sequence A006522. If no three diagonals of a convex polygon are concurrent at a point in the interior, the number of interior intersections of diagonals is given by; this holds, for any regular polygon with an odd number of sides.
The formula follows from the fact that each intersection is uniquely determined by the four endpoints of the two intersecting diagonals: the number of intersections is thus the number of combinations of the n vertices four at a time. A triangle has no diagonals. A square has two diagonals of equal length; the ratio of a diagonal to a side is 2 ≈ 1.414. A regular pentagon has five diagonals all of the same length; the ratio of a diagonal to a side is the golden ratio, 1 + 5 2 ≈ 1.618. A regular hexagon has nine diagonals: the six shorter ones are equal to each other in length; the ratio of a long diagonal to a side is 2. A regular heptagon has 14 diagonals; the seven shorter ones equal each other, the seven longer ones equal each other. The reciprocal of the side equals the sum of the reciprocals of a long diagonal. In any regular n-gon with n the long diagonals all intersect each other at the polygon's center. A polyhedron may have two different types of diagonals: face diagonals on the various faces, connecting non-adjacent vertices on the same face.
Just as a triangle has no diagonals, so a tetrahedron has no face diagonals and no space diagonals. A cuboid has two diagonals on each of four space diagonals. In the case of a square matrix, the main or principal diagonal is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, these would be entries A i j with i = j. For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere: ( 1 0 0 0 1 0
Canadian English
Canadian English is the set of varieties of the English language native to Canada. According to the 2011 census, English was the first language of 19 million Canadians, or 57% of the population. A larger number, 28 million people, reported using English as their dominant language. 82% of Canadians outside the province of Quebec reported speaking English natively, but within Quebec the figure was just 7.7% as most of its residents are native speakers of Quebec French. Canadian English contains major elements of both British English and American English, as well as many uniquely Canadian characteristics. While, broadly speaking, Canadian English tends to be closest to American English in terms of linguistic distance, the precise influence of American English, British English and other sources on Canadian English varieties has been the ongoing focus of systematic studies since the 1950s. Phonologically and American English are classified together as North American English, emphasizing the fact that the vast majority of outsiders other native English speakers, cannot distinguish the typical accents of the two countries by sound alone.
There are minor disagreements over the degree to which Canadians and Americans themselves can differentiate their own two accents, there is evidence that some Western American English is undergoing a vowel shift coinciding with the one first reported in mainland Canadian English in the early 1990s. The term "Canadian English" is first attested in a speech by the Reverend A. Constable Geikie in an address to the Canadian Institute in 1857. Geikie, a Scottish-born Canadian, reflected the Anglocentric attitude that would be prevalent in Canada for the next hundred years when he referred to the language as "a corrupt dialect", in comparison with what he considered the proper English spoken by immigrants from Britain. Canadian English is the product of five waves of immigration and settlement over a period of more than two centuries; the first large wave of permanent English-speaking settlement in Canada, linguistically the most important, was the influx of Loyalists fleeing the American Revolution, chiefly from the Mid-Atlantic States – as such, Canadian English is believed by some scholars to have derived from northern American English.
Canadian English has been developing features of its own since the early 19th century. The second wave from Britain and Ireland was encouraged to settle in Canada after the War of 1812 by the governors of Canada, who were worried about American dominance and influence among its citizens. Further waves of immigration from around the globe peaked in 1910, 1960 and at the present time had a lesser influence, but they did make Canada a multicultural country, ready to accept linguistic change from around the world during the current period of globalization; the languages of Aboriginal peoples in Canada started to influence European languages used in Canada before widespread settlement took place, the French of Lower Canada provided vocabulary, with words such as toque and portage, to the English of Upper Canada. Studies on earlier forms of English in Canada are rare, yet connections with other work to historical linguistics can be forged. An overview of diachronic work on Canadian English, or diachronically-relevant work, is Dollinger.
Until the 2000s all commentators on the history of CanE have argued from the "language-external" history, i.e. social and political history. An exception has been in the area of lexis, where Avis et al's Dictionary of Canadianisms on Historical Principles, offered real-time historical data through its quotations. Historical linguists have started to study earlier Canadian English on historical linguistic data. DCHP-1 is now available in open access. Most notably, Dollinger pioneered the historical corpus linguistic approach for English in Canada with CONTE and offers a developmental scenario for 18th- and 19th-century Ontario. Reuter, with a 19th-century newspaper corpus from Ontario, has confirmed the scenario laid out in Dollinger. Canadian English included a class-based sociolect known as Canadian dainty. Treated as a marker of upper-class prestige in the 19th century and the early part of the 20th, Canadian dainty was marked by the use of some features of British English pronunciation, resulting in an accent similar to the Mid-Atlantic accent known in the United States.
This accent faded in prominence following World War II, when it became stigmatized as pretentious, is now never heard in contemporary Canadian life outside of archival recordings used in film, television or radio documentaries. Canadian spelling of the English language combines American conventions. Words such as realize and paralyze are spelled with -ize or -yze rather than -ise or -yse. French-derived words that in American English end with -or and -er, such as color or center retain British spellings. While the United States uses the Anglo-French spelling defense and offense, most Canadians use the British spellings defence and offence; some nouns, as in British English, take -ice while matching verbs take -ise – for example and licence are nouns while practise and license are the re
Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, deducing many other propositions from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system; the Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute metaphysical, sense.
Today, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, Euclidean space is a good approximation for it only over short distances. Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects; this is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas. The Elements is a systematization of earlier knowledge of geometry, its improvement over earlier treatments was recognized, with the result that there was little interest in preserving the earlier ones, they are now nearly all lost. There are 13 books in the Elements: Books I–IV and VI discuss plane geometry.
Many results about plane figures are proved, for example "In any triangle two angles taken together in any manner are less than two right angles." and the Pythagorean theorem "In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle." Books V and VII–X deal with number theory, with numbers treated geometrically as lengths of line segments or areas of regions. Notions such as prime numbers and rational and irrational numbers are introduced, it is proved. Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base; the platonic solids are constructed. Euclidean geometry is an axiomatic system, in which all theorems are derived from a small number of simple axioms; until the advent of non-Euclidean geometry, these axioms were considered to be true in the physical world, so that all the theorems would be true. However, Euclid's reasoning from assumptions to conclusions remains valid independent of their physical reality.
Near the beginning of the first book of the Elements, Euclid gives five postulates for plane geometry, stated in terms of constructions: Let the following be postulated:To draw a straight line from any point to any point. To produce a finite straight line continuously in a straight line. To describe a circle with any centre and distance; that all right angles are equal to one another.: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles. Although Euclid only explicitly asserts the existence of the constructed objects, in his reasoning they are implicitly assumed to be unique; the Elements include the following five "common notions": Things that are equal to the same thing are equal to one another. If equals are added to equals the wholes are equal. If equals are subtracted from equals the differences are equal.
Things that coincide with one another are equal to one another. The whole is greater than the part. Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more complete sets of axioms. To the ancients, the parallel postulate seemed less obvious than the others, they aspired to create a system of certain propositions, to them it seemed as if the parallel line postulate required proof from simpler statements. It is now known that such a proof is impossible, since one can construct consistent systems of geometry in which the parallel postulate is true, others in which it is false. Euclid himself seems to have considered it as being qualitatively different from the others, as evidenced by the organization of the Elements: his first 28 propositions are those that can be proved without it. Many alternative axioms can be formulated. For example, Playfair's axiom states: In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the giv
