Question: Information: - In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral (a quadrilateral whose vertices lie on a common circle). The theorem is named after the Greek astronomer and mathematician Ptolemy (Claudius Ptolemaeus). Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy. - The Prytanee National Militaire, originally College Royal Henry-Le-Grand or College Henri-IV de La Fleche, is a French school managed by the military, offering regular secondary education as well as special preparatory classes, equivalent in level to the first years of university, for students who wish to enter French military academies. The school is located in western France in the city of LaFleche. At first founded in 1604 by the King Henri IV the school was given to the Jesuits in the aim to "instruct the young people and make it fall in love of sciences, honor and virtue, in order to be able to serve". It then became the "Prytanee" wanted by Napoleon in 1800. - Emile Michel Hyacinthe Lemoine (18401912) was a French civil engineer and a mathematician, a geometer in particular. He was educated at a variety of institutions, including the Prytanee National Militaire and, most notably, the Ecole Polytechnique. Lemoine taught as a private tutor for a short period after his graduation from the latter school. - Alexandria (or ; Arabic: '; '; "") is the second largest city and a major economic centre in Egypt, extending about along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it highly vulnerable to rising sea levels. Alexandria is Egypt's largest seaport, serving approximately 80% of Egypt's imports and exports. It is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination. - John Casey ( 12 May 1820 , Kilbehenny , Ireland -- 3 January 1891 , Dublin ) was a respected Irish geometer . He is most famous for Casey 's theorem on a circle that is tangent to four other circles , an extension of the problem of Apollonius . However , he contributed several novel proofs and perspectives on Euclidean geometry . He and Emile Lemoine are considered to be the co-founders of the modern geometry of the circle and the triangle . - A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. - Euclidean geometry is a mathematical system attributed to the 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, and deducing many other propositions (theorems) 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. - Ulster (or "Cuige Uladh" , Ulster Scots: "Ulster" or "Ulster") is a province in the north of the island of Ireland. In ancient Ireland, it was one of the fifths ruled by a "ri ruirech", or "king of over-kings". - Ireland (Ulster-Scots: ) is an island in the North Atlantic. It is separated from Great Britain to its east by the North Channel, the Irish Sea, and St George's Channel. Ireland is the second-largest island of the British Isles, the third-largest in Europe, and the twentieth-largest on Earth. - In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system. - The River Liffey (Irish: "An Life") is a river in Ireland, which flows through the centre of Dublin. Its major tributaries include the River Dodder, the River Poddle and the River Camac. The river supplies much of Dublin's water and a range of recreational opportunities. - A civil engineer is a person who practices civil engineering the application of planning, designing, constructing, maintaining, and operating infrastructures while protecting the public and environmental health, as well as improving existing infrastructures that have been neglected. - Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. - The British Isles are a group of islands off the north-western coast of continental Europe that consist of the islands of Great Britain, Ireland and over six thousand smaller isles. Situated in the North Atlantic, the islands have a total area of approximately 315,159km, and a combined population of just under 70 million. Two sovereign states are located on the islands: Ireland (which covers roughly five-sixths of the island with the same name) and the United Kingdom of Great Britain and Northern Ireland. The British Isles also include three Crown Dependencies: the Isle of Man and, by tradition, the Bailiwick of Jersey and the Bailiwick of Guernsey in the Channel Islands, although the latter are not physically a part of the archipelago. - In Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure1). Apollonius of Perga (ca. 262 190 BCE) posed and solved this famous problem in his work ("", "Tangencies"); this work has been lost, but a 4th-century report of his results by Pappus of Alexandria has survived. Three given circles generically have eight different circles that are tangent to them (Figure2) and each solution circle encloses or excludes the three given circles in a different way: in each solution, a different subset of the three circles is enclosed (its complement is excluded) and there are 8 subsets of a set whose cardinality is 3, since 8=2. - Pappus of Alexandria (c. 290 c. 350 AD) was one of the last great Alexandrian mathematicians of Antiquity, known for his "Synagoge" () or "Collection" (c. 340), and for Pappus's hexagon theorem in projective geometry. Nothing is known of his life, other than, (from his own writings) that he had a son named Hermodorus, and was a teacher in Alexandria. - The Greater Dublin Area (GDA) (Irish: "Morcheantar Bhaile Atha Cliath"), or simply Greater Dublin is the city of Dublin and its hinterland, with varying definitions as to its extent. The National Transport Authority has defined the Greater Dublin Area as including the counties of Dublin (Dublin City, South Dublin, Dun LaoghaireRathdown and Fingal), Meath, Kildare, and Wicklow. However, this area does not comprise a formal political unit. The term can also apply more narrowly to the Dublin urban area and nearby suburban towns. The general area is sometimes referred to colloquially as The Pale. - In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey. - Apollonius of Perga (c. 262 c. 190 BC) was a Greek geometer and astronomer noted for his writings on conic sections. His innovative methodology and terminology, especially in the field of conics, influenced many later scholars including Ptolemy, Francesco Maurolico, Johannes Kepler, Isaac Newton, and Rene Descartes. Apollonius gave the ellipse, the parabola, and the hyperbola their modern names. The hypothesis of eccentric orbits, or equivalently, deferent and epicycles, to explain the apparent motion of the planets and the varying speed of the Moon, is also attributed to him. Ptolemy describes Apollonius' theorem in the "Almagest" XII.1. The crater Apollonius on the Moon is named in his honor. - Geometry (from the ; "geo-" "earth", "-metron" "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. - A tutor is an instructor who gives private lessons. Shadow education is a name for private supplementary tutoring that is offered outside the mainstream education system. - Euclid ("Eukleids" ; fl. 300 BCE), sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of PtolemyI (323283 BCE). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. - Dublin is the capital and largest city of Ireland. Dublin is in the province of Leinster on Ireland's east coast, at the mouth of the River Liffey. The city has an urban area population of 1,345,402. The population of the Greater Dublin Area, , was 1,904,806 people. - Ecole Polytechnique (also known by the nickname "X") is a French public institution of higher education and research in Palaiseau near Paris. 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Answer:
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