BisectionIn geometry, bisection is the division of something into two equal or congruent parts (having the same shape and size). Usually it involves a bisecting line, also called a 'bisector'. The most often considered types of bisectors are the 'segment bisector' (a line that passes through the midpoint of a given segment) and the 'angle bisector' (a line that passes through the apex of an angle, that divides it into two equal angles). In three-dimensional space, bisection is usually done by a bisecting plane, also called the 'bisector'.
Proof by contradictionIn logic, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Although it is quite freely used in mathematical proofs, not every school of mathematical thought accepts this kind of nonconstructive proof as universally valid. More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved.
Cross-ratioIn geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points A, B, C, D on a line, their cross ratio is defined as where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.
Constructive proofIn mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an effective proof.
Koine GreekKoine Greek (UKˈkɔɪni ; USˈkɔɪneɪ , kɔɪˈneɪ ; Koine hē koinè diálektos), also known as Hellenistic Greek, common Attic, the Alexandrian dialect, Biblical Greek or New Testament Greek, was the common supra-regional form of Greek spoken and written during the Hellenistic period, the Roman Empire and the early Byzantine Empire. It evolved from the spread of Greek following the conquests of Alexander the Great in the fourth century BC, and served as the lingua franca of much of the Mediterranean region and the Middle East during the following centuries.
Greek alphabetThe Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BC. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as well as consonants. In Archaic and early Classical times, the Greek alphabet existed in many local variants, but, by the end of the 4th century BC, the Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard and it is this version that is still used for Greek writing today.
Line (geometry)In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist embedded in two, three, or higher dimensional spaces. The word line may also refer to a line segment in everyday life that has two points to denote its ends (endpoints). A line can be referred to by two points that lie on it (e.g. ) or by a single letter (e.g. ).
Greek languageGreek (Elliniká, eliniˈka; Hellēnikḗ) is an independent branch of the Indo-European family of languages, native to Greece, Cyprus, Italy (in Calabria and Salento), southern Albania, and other regions of the Balkans, the Black Sea coast, Asia Minor, and the Eastern Mediterranean. It has the longest documented history of any Indo-European language, spanning at least 3,400 years of written records. Its writing system is the Greek alphabet, which has been used for approximately 2,800 years; previously, Greek was recorded in writing systems such as Linear B and the Cypriot syllabary.
Romanization of GreekRomanization of Greek is the transliteration (letter-mapping) and/or transcription (sound-mapping) of text from the Greek alphabet into the Latin alphabet. The conventions for writing and romanizing Ancient Greek and Modern Greek differ markedly. The sound of the English letter B (/b/) was written as β in ancient Greek but is now written as the digraph μπ, while the modern β sounds like the English letter V (/v/) instead.
Great BritainGreat Britain is an island in the North Atlantic Ocean off the north-west coast of continental Europe, consisting of England, Scotland and Wales. With an area of , it is the largest of the British Isles, the largest European island and the ninth-largest island in the world. It is dominated by a maritime climate with narrow temperature differences between seasons. The island of Ireland, with an area 40 per cent that of Great Britain, is to the west—these islands, along with over 1,000 smaller surrounding islands and named substantial rocks, form the British Isles archipelago.
