112-511: (Redirected from The Triangle ) [REDACTED] Look up Triangle , triangle , scalene triangle , or triangular in Wiktionary, the free dictionary. A triangle is a geometric shape with three sides. Triangle may also refer to: Mathematics [ edit ] Exact triangle , a collection of objects in category theory Triangle inequality , Euclid's proposition that
224-415: A . {\displaystyle q_{a}={\frac {2Ta}{a^{2}+2T}}={\frac {ah_{a}}{a+h_{a}}}.} The largest possible ratio of the area of the inscribed square to the area of the triangle is 1/2, which occurs when a 2 = 2 T {\displaystyle a^{2}=2T} , q = a / 2 {\displaystyle q=a/2} , and the altitude of the triangle from
336-404: A {\displaystyle a} , h a {\displaystyle h_{a}} from the side a {\displaystyle a} , and the triangle's area T {\displaystyle T} are related according to q a = 2 T a a 2 + 2 T = a h a a + h
448-441: A ) ( s − b ) ( s − c ) . {\displaystyle T={\sqrt {s(s-a)(s-b)(s-c)}}.} Because the ratios between areas of shapes in the same plane are preserved by affine transformations , the relative areas of triangles in any affine plane can be defined without reference to a notion of distance or squares. In any affine space (including Euclidean planes), every triangle with
560-562: A British South American Airways (BSAA) passenger airplane; the March 1948 disappearance of a fishing skiff with three men, including jockey Albert Snider ; the December 1948 disappearance of an Airborne Transport DC-3 charter flight en route from Puerto Rico to Miami; and the January 1949 disappearance of Star Ariel , another BSAA passenger airplane. Flight 19 was covered again in
672-454: A pseudotriangle . A pseudotriangle is a simply-connected subset of the plane lying between three mutually tangent convex regions. These sides are three smoothed curved lines connecting their endpoints called the cusp points . Any pseudotriangle can be partitioned into many pseudotriangles with the boundaries of convex disks and bitangent lines , a process known as pseudo-triangulation. For n {\displaystyle n} disks in
784-582: A 1953 film starring Douglas Fairbanks Jr. Television [ edit ] The Triangle (miniseries) , a 2005 Sci-Fi Channel series Triangle (1981 TV series) , a 1980s BBC soap opera Triangle (2014 TV series) , a 2014 MBC Korean drama "Triangle" ( Buffy the Vampire Slayer ) , 2001 "Triangle" ( The X-Files ) , 1998 "The Triangle" ( The Amazing World of Gumball ) , 2015 "Triangles", an episode of Private Practice Places [ edit ] Bermuda Triangle , area of
896-423: A 2011 EP by 10,000 Maniacs Film [ edit ] Triangle Film Corporation , a film studio in the U.S. during the silent era The Triangle (film) , a 2001 made-for-TV thriller Triangle (2007 film) , a Hong Kong crime-thriller Triangle (2009 British film) , a British-Australian psychological thriller Triangle (2009 South Korean film) , a South Korean-Japanese comedy The Triangle ,
1008-472: A North American Wye rail Triangle Fraternity , social fraternity for STEM Students Triangle offense , an offensive strategy used in basketball Triangle Rewards, a loyalty program offered by Canadian Tire The Triangle (newspaper) , at Drexel University The Triangle, Manchester , a building in England Triangles (novel) , a 2011 novel by Ellen Hopkins Delta (letter) , in
1120-398: A circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a unique Steiner circumellipse , which passes through the triangle's vertices and has its center at the triangle's centroid. Of all ellipses going through the triangle's vertices, it has the smallest area. The Kiepert hyperbola
1232-521: A circular triangle whose sides are all convex. An example of a circular triangle with three convex edges is a Reuleaux triangle , which can be made by intersecting three circles of equal size. The construction may be performed with a compass alone without needing a straightedge, by the Mohr–Mascheroni theorem . Alternatively, it can be constructed by rounding the sides of an equilateral triangle. A special case of concave circular triangle can be seen in
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#17328018781191344-508: A corresponding triangle in a model space like hyperbolic or elliptic space. For example, a CAT(k) space is characterized by such comparisons. Bermuda Triangle The Bermuda Triangle , also known as the Devil's Triangle , is a loosely defined region between Florida , Bermuda , and Puerto Rico in the southwestern North Atlantic Ocean where a number of aircraft and ships have disappeared under mysterious circumstances. The idea of
1456-457: A new concept of trigonometric functions . The primary trigonometric functions are sine and cosine , as well as the other functions. They can be defined as the ratio between any two sides of a right triangle . In a scalene triangle, the trigonometric functions can be used to find the unknown measure of either a side or an internal angle; methods for doing so use the law of sines and the law of cosines . Any three angles that add to 180° can be
1568-409: 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
1680-578: A pattern of strange events in the region, dating back to at least 1840. The next year, Gaddis expanded this article into a book, Invisible Horizons . Other writers elaborated on Gaddis' ideas, including John Wallace Spencer ( Limbo of the Lost , 1969, repr. 1973); Charles Berlitz ( The Bermuda Triangle , 1974) ; and Richard Winer ( The Devil's Triangle , 1974). Various of these authors incorporated supernatural elements. Sand's article in Fate described
1792-523: A port with the same name in the Pacific Ocean. Kusche also argued that a large percentage of the incidents that sparked allegations of the Triangle's mysterious influence actually occurred well outside it. Often his research was simple: he would review period newspapers of the dates of reported incidents and find reports on possibly relevant events, like unusual weather, that were never mentioned in
1904-580: A pseudotriangle, the partition gives 2 n − 2 {\displaystyle 2n-2} pseudotriangles and 3 n − 3 {\displaystyle 3n-3} bitangent lines. The convex hull of any pseudotriangle is a triangle. A non-planar triangle is a triangle not included in Euclidean space , roughly speaking a flat space. This means triangles may also be discovered in several spaces, as in hyperbolic space and spherical geometry . A triangle in hyperbolic space
2016-464: A reference triangle, the nearest points on the three sides serve as the vertices of the pedal triangle of that point. If the interior point is the circumcenter of the reference triangle, the vertices of the pedal triangle are the midpoints of the reference triangle's sides, and so the pedal triangle is called the midpoint triangle or medial triangle. The midpoint triangle subdivides the reference triangle into four congruent triangles which are similar to
2128-537: A region of North Carolina, U.S., anchored by Raleigh, Durham, and Chapel Hill Triangle, New York , United States Triangle, Virginia , United States Triangle, West Yorkshire , a village in Calderdale, England Triangle, Zimbabwe Other uses [ edit ] Triangle (chart pattern) , in financial technical analysis Triangle (novel) , a 1983 Star Trek novel Triangle (Paris building) Triangle (railway) , an English term equivalent to
2240-409: A right angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter ; this point is the center of the circumcircle , the circle passing through all three vertices. Thales' theorem implies that if the circumcenter is located on the side of the triangle, then the angle opposite that side is a right angle. If the circumcenter is located inside the triangle, then
2352-403: A short article by George X. Sand that was the first to lay out the now-familiar triangular area where the losses took place. Sand recounted the loss of several planes and ships since World War II: the disappearance of Sandra , a tramp steamer ; the December 1945 loss of Flight 19 , a group of five US Navy torpedo bombers on a training mission; the January 1948 disappearance of Star Tiger ,
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#17328018781192464-434: A similar triangle: As discussed above, every triangle has a unique inscribed circle (incircle) that is interior to the triangle and tangent to all three sides. Every triangle has a unique Steiner inellipse which is interior to the triangle and tangent at the midpoints of the sides. Marden's theorem shows how to find the foci of this ellipse . This ellipse has the greatest area of any ellipse tangent to all three sides of
2576-400: A simple polygon has a relationship to the ear , a vertex connected by two other vertices, the diagonal between which lies entirely within the polygon. The two ears theorem states that every simple polygon that is not itself a triangle has at least two ears. One way to identify locations of points in (or outside) a triangle is to place the triangle in an arbitrary location and orientation in
2688-494: A single point, the symmedian point of the triangle. The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees. This fact is equivalent to Euclid's parallel postulate . This allows the determination of the measure of the third angle of any triangle, given the measure of two angles. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary ) to an interior angle. The measure of an exterior angle of
2800-413: A square's vertices lie on a side of the triangle, so two of them lie on the same side and hence one side of the square coincides with part of a side of the triangle). In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of
2912-439: A time/space warp that sucks the objects around it into a parallel universe. Others attribute the events to UFOs . Charles Berlitz , author of various books on anomalous phenomena, lists several theories attributing the losses in the Triangle to anomalous or unexplained forces. Compass problems are one of the cited phrases in many Triangle incidents. While some have theorized that unusual local magnetic anomalies may exist in
3024-408: A triangle are often constructed by proving that three symmetrically constructed points are collinear ; here Menelaus' theorem gives a useful general criterion. In this section, just a few of the most commonly encountered constructions are explained. A perpendicular bisector of a side of a triangle is a straight line passing through the midpoint of the side and being perpendicular to it, forming
3136-400: A triangle is equal to the sum of the measures of the two interior angles that are not adjacent to 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, and indeed, this is true for any convex polygon, no matter how many sides it has. Another relation between the internal angles and triangles creates
3248-406: A triangle of area at most equal to 2 T {\displaystyle 2T} . Equality holds only if the polygon is a parallelogram . The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. As mentioned above, every triangle has a unique circumcircle,
3360-587: A triangle, for instance, a spherical triangle or hyperbolic triangle . A geodesic triangle is a region of a general two-dimensional surface enclosed by three sides that are straight relative to the surface ( geodesics ). A curvilinear triangle is a shape with three curved sides, for instance, a circular triangle with circular-arc sides. This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted. Triangles are classified into different types based on their angles and
3472-502: Is a cyclic hexagon with vertices given by the six intersections of the sides of a triangle with the three lines that are parallel to the sides and that pass through its symmedian point . In either its simple form or its self-intersecting form , the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. Every convex polygon with area T {\displaystyle T} can be inscribed in
Triangle (disambiguation) - Misplaced Pages Continue
3584-431: Is a formula for finding the area of a triangle from the lengths of its sides a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} . Letting s = 1 2 ( a + b + c ) {\displaystyle s={\tfrac {1}{2}}(a+b+c)} be the semiperimeter , T = s ( s −
3696-531: Is a river within an ocean, and, like a river, it can and does carry floating objects. It has a maximum surface velocity of about 2 m/s (6.6 ft/s). A small plane making a water landing or a boat having engine trouble can be carried away from its reported position by the current. One of the most cited explanations in official inquiries as to the loss of any aircraft or vessel is human error . Human stubbornness may have caused businessman Harvey Conover to lose his sailing yacht, Revonoc , as he sailed into
3808-495: Is a solid whose boundary is covered by flat polygonals known as the faces, sharp corners known as the vertices, and line segments known as the edges. Polyhedra in some cases can be classified, judging from the shape of their faces. For example, when polyhedra have all equilateral triangles as their faces, they are known as deltahedra . Antiprisms have alternating triangles on their sides. Pyramids and bipyramids are polyhedra with polygonal bases and triangles for lateral faces;
3920-420: Is called a hyperbolic triangle , and it can be obtained by drawing on a negatively curved surface, such as a saddle surface . Likewise, a triangle in spherical geometry is called a spherical triangle , and it can be obtained by drawing on a positively curved surface such as a sphere . The triangles in both spaces have properties different from the triangles in Euclidean space. For example, as mentioned above,
4032-461: Is considered the most likely cause of sinking. Carroll A. Deering , a five-masted schooner built in 1919, was found hard aground and abandoned at Diamond Shoals , near Cape Hatteras , North Carolina , on 31 January 1921. FBI investigation into the Deering scrutinized, then ruled out, multiple theories as to why and how the ship was abandoned, including piracy, domestic Communist sabotage and
4144-444: Is different from Wikidata All article disambiguation pages All disambiguation pages 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
4256-464: Is more than two thousand years old, having been defined in Book One of Euclid's Elements . The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations. Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle , a triangle with two sides having
4368-401: Is not located on Euler's line. A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid or geometric barycenter. The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its center of mass :
4480-564: Is required, usually through a library connected to a college or university. The following websites have either online material that supports the popular version of the Bermuda Triangle, or documents published from official sources as part of hearings or inquiries, such as those conducted by the United States Navy or United States Coast Guard. Copies of some inquiries are not online and may have to be ordered; for example,
4592-402: Is the matrix determinant . The triangle inequality states that the sum of the lengths of any two sides of a triangle must be greater than or equal to the length of the third side. Conversely, some triangle with three given positive side lengths exists if and only if those side lengths satisfy the triangle inequality. The sum of two side lengths can equal the length of the third side only in
Triangle (disambiguation) - Misplaced Pages Continue
4704-408: Is the center of the triangle's incircle . The incircle is the circle that lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the excircles ; they lie outside the triangle and touch one side, as well as the extensions of the other two. The centers of the incircles and excircles form an orthocentric system . The midpoints of
4816-412: Is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. The orthocenter lies inside the triangle if and only if the triangle is acute. An angle bisector of a triangle is a straight line through a vertex that cuts the corresponding angle in half. The three angle bisectors intersect in a single point, the incenter , which
4928-496: Is unique conic that passes through the triangle's three vertices, its centroid, and its circumcenter. Of all triangles contained in a given convex polygon , one with maximal area can be found in linear time; its vertices may be chosen as three of the vertices of the given polygon. A circular triangle is a triangle with circular arc edges. The edges of a circular triangle may be either convex (bending outward) or concave (bending inward). The intersection of three disks forms
5040-651: Is why engineering makes use of tetrahedral trusses . Triangulation means the partition of any planar object into a collection of triangles. For example, in polygon triangulation , a polygon is subdivided into multiple triangles that are attached edge-to-edge, with the property that their vertices coincide with the set of vertices of the polygon. In the case of a simple polygon with n {\displaystyle n} sides, there are n − 2 {\displaystyle n-2} triangles that are separated by n − 3 {\displaystyle n-3} diagonals. Triangulation of
5152-498: The Cartesian plane , and to use Cartesian coordinates. While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane. Two systems avoid that feature, so that the coordinates of a point are not affected by moving the triangle, rotating it, or reflecting it as in a mirror, any of which gives a congruent triangle, or even by rescaling it to
5264-570: The Irish coast ," according to a 1977 BBC program. Consequently, the determination of which accidents occurred inside the triangle depends on which writer reported them. Larry Kusche , author of The Bermuda Triangle Mystery: Solved (1975), argued that many claims of Gaddis and subsequent writers were exaggerated, dubious or unverifiable. Kusche's research revealed a number of inaccuracies and inconsistencies between Berlitz's accounts and statements from eyewitnesses, participants, and others involved in
5376-483: The Royal Naval Dockyard, Bermuda for Falmouth , England on 31 January 1880. It was presumed that she sank in a powerful storm which crossed her route a couple of weeks after she sailed, and that her crew being composed primarily of inexperienced trainees may have been a contributing factor. The search for evidence of her fate attracted worldwide attention at the time (connection is also often made to
5488-583: The USGS describe large stores of undersea hydrates worldwide, including the Blake Ridge area, off the coast of the southeastern United States. However, according to the USGS, no large releases of gas hydrates are believed to have occurred in the Bermuda Triangle for the past 15,000 years. The sail training ship HMS Atalanta (originally named HMS Juno ) disappeared with her entire crew after setting sail from
5600-487: The simplicial polytopes . Each triangle has many special points inside it, on its edges, or otherwise associated with it. They are constructed by finding three lines associated symmetrically with the three sides (or vertices) and then proving that the three lines meet in a single point. An important tool for proving the existence of these points is Ceva's theorem , which gives a criterion for determining when three such lines are concurrent . Similarly, lines associated with
5712-534: The 1878 loss of the training ship HMS Eurydice , which foundered after departing the Royal Naval Dockyard in Bermuda for Portsmouth on 6 March), and she was alleged decades later to have been a victim of the mysterious triangle, an allegation resoundingly refuted by the research of author David Francis Raine in 1997. The incident resulting in the single largest loss of life in the history of
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#17328018781195824-483: The 1972 explosion and sinking of the tanker V. A. Fogg , the Coast Guard photographed the wreck and recovered several bodies, in contrast with one Triangle author's claim that all the bodies had vanished, with the exception of the captain, who was found sitting in his cabin at his desk, clutching a coffee cup. In addition, V. A. Fogg sank off the coast of Texas , nowhere near the commonly accepted boundaries of
5936-517: The April 1962 issue of The American Legion Magazine . In it, author Allan W. Eckert wrote that the flight leader had been heard saying, "We cannot be sure of any direction ... everything is wrong ... strange ... the ocean doesn't look as it should." In February 1964, Vincent Gaddis wrote an article called "The Deadly Bermuda Triangle" in Argosy saying Flight 19 and other disappearances were part of
6048-545: The Bahamas, which is in the Triangle by some definitions. Followers of the purported psychic Edgar Cayce take his prediction that evidence of Atlantis would be found in 1968 as referring to the discovery of the Bimini Road. Believers describe the formation as a road, wall, or other structure, but the Bimini Road is of natural origin. Some hypothesize that a parallel universe exists in the Bermuda Triangle region, causing
6160-524: The Bermuda Triangle as a real phenomenon have offered a number of explanatory approaches. Triangle writers have used a number of supernatural concepts to explain the events. One explanation pins the blame on leftover technology from the mythical lost continent of Atlantis . Sometimes connected to the Atlantis story is the submerged rock formation known as the Bimini Road off the island of Bimini in
6272-409: The Bermuda Triangle that it could be very difficult to locate an aircraft lost at sea due to the vast search area, and although the disappearance might be mysterious, that did not make it paranormal or unexplainable. Radford further noted the importance of double-checking information as the mystery surrounding the Bermuda Triangle had been created by people who had neglected to do so. Persons accepting
6384-399: The Bermuda Triangle, Winer quoted from a letter he had received from Mr J.E. Challenor of Barbados: On the morning of September 22, Connemara IV was lying to a heavy mooring in the open roadstead of Carlisle Bay . Because of the approaching hurricane, the owner strengthened the mooring ropes and put out two additional anchors. There was little else he could do, as the exposed mooring was
6496-618: The Greek alphabet, whose uppercase resembles a triangle (Δ) Trigonodes hyppasia or Triangles, a species of moth See also [ edit ] Triangle Lake (disambiguation) Triangle Park (disambiguation) All pages with titles containing Triangle Triangeln station , a railway station in Malmö, Sweden Triangular trade Triangular election in France Triangulum (disambiguation) Topics referred to by
6608-676: The North Atlantic between Bermuda, Florida, and Puerto Rico Le Triangle , a residential district in Montreal Triangle, Newfoundland and Labrador , Canada Triangle (Israel) , a concentration of Israeli Arab towns Triangle Region (Denmark) ( Trekanten ), a sub-region on the Jutland Peninsula Trianglen, Copenhagen , a large intersection in Copenhagen, Denmark Research Triangle ,
6720-402: The Triangle. The Nova / Horizon episode The Case of the Bermuda Triangle , aired on 27 June 1976, was highly critical, stating that "When we've gone back to the original sources or the people involved, the mystery evaporates. Science does not have to answer questions about the Triangle because those questions are not valid in the first place ... Ships and planes behave in the Triangle
6832-676: The Triangle. Many Atlantic hurricanes pass through the Triangle as they recurve off the Eastern Seaboard, and, before the advent of weather satellites , ships often had little to no warning of a hurricane's approach. A powerful downdraft of cold air was suspected to be a cause in the sinking of Pride of Baltimore on 14 May 1986. The crew of the sunken vessel noted the wind suddenly shifted and increased velocity from 32 km/h (20 mph) to 97–145 km/h (60–90 mph). A National Hurricane Center satellite specialist, James Lushine, stated "during very unstable weather conditions
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#17328018781196944-498: The US Navy not related to combat occurred when the collier Cyclops , carrying a full load of manganese ore and with one engine out of action, went missing without a trace with a crew of 306 sometime after 4 March 1918, after departing the island of Barbados . Although there is no strong evidence for any single theory, many independent theories exist, some blaming storms, some capsizing, and some suggesting that wartime enemy activity
7056-420: The altitude can be calculated using trigonometry, h = a sin ( γ ) {\displaystyle h=a\sin(\gamma )} , so the area of the triangle is: T = 1 2 a b sin γ . {\displaystyle T={\tfrac {1}{2}}ab\sin \gamma .} Heron's formula , named after Heron of Alexandria ,
7168-828: The angles of a triangle can also be stated using trigonometric functions. For example, a triangle with angles α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } exists if and only if cos 2 α + cos 2 β + cos 2 γ + 2 cos ( α ) cos ( β ) cos ( γ ) = 1. {\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma +2\cos(\alpha )\cos(\beta )\cos(\gamma )=1.} Two triangles are said to be similar , if every angle of one triangle has
7280-419: The angles of a triangle on a sphere is 180 ∘ × ( 1 + 4 f ) {\displaystyle 180^{\circ }\times (1+4f)} , where f {\displaystyle f} is the fraction of the sphere's area enclosed by the triangle. In more general spaces, there are comparison theorems relating the properties of a triangle in the space to properties of
7392-511: The area as "a watery triangle bounded roughly by Florida , Bermuda and Puerto Rico ". The Argosy article by Gaddis further delineated the boundaries, giving its vertices as Miami , San Juan , and Bermuda. Subsequent writers did not necessarily follow this definition. Some writers gave different boundaries and vertices to the triangle, with the total area varying from 1.3 to 3.9 million km (0.50 to 1.51 million sq mi). "Indeed, some writers even stretch it as far as
7504-623: The area as uniquely prone to disappearances arose in the mid-20th century, but most reputable sources dismiss the idea that there is any mystery. The earliest suggestion of unusual disappearances in the Bermuda area appeared in an article written by Edward Van Winkle Jones of the Miami Herald that was distributed by the Associated Press and appeared in various American newspapers on 17 September 1950. Two years later, Fate magazine published "Sea Mystery at Our Back Door",
7616-438: The area of an arbitrary triangle. One of the oldest and simplest is to take half the product of the length of one side b {\displaystyle b} (the base) times the corresponding altitude h {\displaystyle h} : T = 1 2 b h . {\displaystyle T={\tfrac {1}{2}}bh.} This formula can be proven by cutting up
7728-535: The area, such anomalies have not been found. Compasses have natural magnetic variations in relation to the magnetic poles , a fact that navigators have known for centuries. Magnetic (compass) north and geographic (true) north are exactly the same only for a small number of places – for example, as of 2000 , in the United States, only those places on a line running from Wisconsin to the Gulf of Mexico . But
7840-425: The base of length a {\displaystyle a} is equal to a {\displaystyle a} . The smallest possible ratio of the side of one inscribed square to the side of another in the same non-obtuse triangle is 2 2 / 3 {\displaystyle 2{\sqrt {2}}/3} . Both of these extreme cases occur for the isosceles right triangle. The Lemoine hexagon
7952-422: The case of a degenerate triangle , one with collinear vertices. Unlike a rectangle, which may collapse into a parallelogram from pressure to one of its points, triangles are sturdy because specifying the lengths of all three sides determines the angles. Therefore, a triangle will not change shape unless its sides are bent or extended or broken or if its joints break; in essence, each of the three sides supports
8064-407: The center of the nine-point circle (red), the centroid (orange), and the circumcenter (green) all lie on a single line, known as Euler's line (red line). The center of the nine-point circle lies at the midpoint between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half that between the centroid and the orthocenter. Generally, the incircle's center
8176-521: The density of the water, and any wreckage would be deposited on the ocean floor or rapidly dispersed by the Gulf Stream. It has been hypothesized that periodic methane eruptions (sometimes called " mud volcanoes ") may produce regions of frothy water that are no longer capable of providing adequate buoyancy for ships. If this were the case, such an area forming around a ship could cause it to sink very rapidly and without warning. Publications by
8288-661: The disappearance stories. Kusche concluded: When the British Channel 4 television program The Bermuda Triangle (1992) was being produced by John Simmons of Geofilms for the Equinox series, the marine insurance market Lloyd's of London was asked if an unusually large number of ships had sunk in the Bermuda Triangle area. Lloyd's determined that large numbers of ships had not sunk there. Lloyd's does not charge higher rates for passing through this area. United States Coast Guard records confirm their conclusion. In fact,
8400-493: The disappearance. A pleasure yacht was found adrift in the Atlantic south of Bermuda on 26 September 1955; it is usually stated in the stories (Berlitz, Winer) that the crew vanished while the yacht survived being at sea during three hurricanes. The 1955 Atlantic hurricane season shows Hurricane Ione passing nearby between 14 and 18 September, with Bermuda being affected by winds of almost gale force. In his second book on
8512-483: The downburst of cold air from aloft can hit the surface like a bomb, exploding outward like a giant squall line of wind and water." An explanation for some of the disappearances has focused on the presence of large fields of methane hydrates (a form of natural gas) on the continental shelves . Laboratory experiments carried out in Australia have proven that bubbles can, indeed, sink a scale model ship by decreasing
8624-660: The end of the incident. According to contemporaneous sources, the Mariner had a history of explosions due to vapor leaks when heavily loaded with fuel, as it might have been for a potentially long search-and-rescue operation. G-AHNP Star Tiger disappeared on 30 January 1948, on a flight from the Azores to Bermuda; G-AGRE Star Ariel disappeared on 17 January 1949, on a flight from Bermuda to Kingston, Jamaica . Both were Avro Tudor IV passenger aircraft operated by British South American Airways . Both planes were operating at
8736-405: The initial incidents. Kusche noted cases where pertinent information went unreported, such as the disappearance of round-the-world yachtsman Donald Crowhurst , which Berlitz had presented as a mystery, despite clear evidence to the contrary. Another example was the ore-carrier recounted by Berlitz as lost without trace three days out of an Atlantic port when in fact it had been lost three days out of
8848-409: The internal angles of a triangle in Euclidean space always add up to 180°. However, the sum of the internal angles of a hyperbolic triangle is less than 180°, and for any spherical triangle, the sum is more than 180°. In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°. By Girard's theorem , the sum of
8960-576: The internal angles of a triangle. Infinitely many triangles have the same angles, since specifying the angles of a triangle does not determine its size. (A degenerate triangle , whose vertices are collinear , has internal angles of 0° and 180°; whether such a shape counts as a triangle is a matter of convention. ) The conditions for three angles α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } , each of them between 0° and 180°, to be
9072-457: The involvement of rum-runners . Flight 19 was a training flight of five TBM Avenger torpedo bombers that disappeared on 5 December 1945, while over the Atlantic. The squadron's flight plan was scheduled to take them due east from Fort Lauderdale for 141 mi (227 km), north for 73 mi (117 km), and then back over a final 140 mi (225 km) leg to complete the exercise. The flight never returned to base. The disappearance
9184-588: The isosceles triangles may be found in the shape of gables and pediments , and the equilateral triangle can be found in the yield sign. The faces of the Great Pyramid of Giza are sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead. Other appearances are in heraldic symbols as in the flag of Saint Lucia and flag of the Philippines . Triangles also appear in three-dimensional objects. A polyhedron
9296-415: The lengths of their sides. Relations between angles and side lengths are a major focus of trigonometry . In particular, the sine, cosine, and tangent functions relate side lengths and angles in right triangles . A triangle is a figure consisting of three line segments, each of whose endpoints are connected. This forms a polygon with three sides and three angles. The terminology for categorizing triangles
9408-555: The market is biased in favor of books, TV specials, and other media that support the Triangle mystery, and against well-researched material if it espouses a skeptical viewpoint. In a 2013 study, the World Wide Fund for Nature identified the world's 10 most dangerous waters for shipping, but the Bermuda Triangle was not among them. Benjamin Radford , an author and scientific paranormal investigator, noted in an interview on
9520-401: The number of supposed disappearances is relatively insignificant considering the number of ships and aircraft that pass through on a regular basis. The Coast Guard is also officially skeptical of the Triangle, noting that they collect and publish, through their inquiries, much documentation contradicting many of the incidents written about by the Triangle authors. In one such incident involving
9632-407: The object can be balanced on its centroid in a uniform gravitational field. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side. If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian . The three symmedians intersect in
9744-488: The official documentation, come from the following works. Some incidents mentioned as having taken place within the Triangle are found only in these sources: ProQuest has newspaper source material for many incidents, archived in Portable Document Format (PDF). The newspapers include The New York Times , The Washington Post , and The Atlanta Constitution . To access this website, registration
9856-453: The only available anchorage. ... In Carlisle Bay, the sea in the wake of Hurricane Janet was awe-inspiring and dangerous. The owner of Connemara IV observed that she had disappeared. An investigation revealed that she had dragged her moorings and gone to sea. On 28 August 1963, a pair of US Air Force KC-135 Stratotanker aircraft collided and crashed into the Atlantic 300 miles (480 km) west of Bermuda. Some writers say that while
9968-411: The other two. A rectangle, in contrast, is more dependent on the strength of its joints in a structural sense. Triangles are strong in terms of rigidity, but while packed in a tessellating arrangement triangles are not as strong as hexagons under compression (hence the prevalence of hexagonal forms in nature ). Tessellated triangles still maintain superior strength for cantilevering , however, which
10080-470: The percussion family Tri Angle (record label) , in New York and London Triangle (band) a Japanese pop group in 1970s The Triangles , Australian band Albums [ edit ] Tri-Angle , a 2004 album by TVXQ Triangle (The Beau Brummels album) , 1967 Triangle (Perfume album) , 2009 Triangle (Diaura album) , 2014 Triangle , a 2008 album by Mi Lu Bing Triangle ,
10192-485: 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 within a unique flat plane . More generally, four points in three-dimensional Euclidean space determine a tetrahedron . In non-Euclidean geometries , three "straight" segments (having zero curvature ) also determine
10304-514: The public may not be as informed, and think there is something mysterious about a compass "changing" across an area as large as the Triangle, which it naturally will. The Gulf Stream ( Florida Current ) is a major surface current, primarily driven by thermohaline circulation that originates in the Gulf of Mexico and then flows through the Straits of Florida into the North Atlantic. In essence, it
10416-434: The reference triangle. The intouch triangle of a reference triangle has its vertices at the three points of tangency of the reference triangle's sides with its incircle. The extouch triangle of a reference triangle has its vertices at the points of tangency of the reference triangle's excircles with its sides (not extended). Every acute triangle has three inscribed squares (squares in its interior such that all four of
10528-772: The same base and oriented area has its apex (the third vertex) on a line parallel to the base, and their common area is half of that of a parallelogram with the same base whose opposite side lies on the parallel line. This affine approach was developed in Book 1 of Euclid's Elements . Given affine coordinates (such as Cartesian coordinates ) ( x A , y A ) {\displaystyle (x_{A},y_{A})} , ( x B , y B ) {\displaystyle (x_{B},y_{B})} , ( x C , y C ) {\displaystyle (x_{C},y_{C})} for
10640-516: The same length is an isosceles triangle , and a triangle with three different-length sides is a scalene triangle . A triangle in which one of the angles is a right angle is a right triangle , a triangle in which all of its angles are less than that angle is an acute triangle , and a triangle in which one of it angles is greater than that angle is an obtuse triangle . These definitions date back at least to Euclid . All types of triangles are commonly found in real life. In man-made construction,
10752-412: The same length. This is a total of six equalities, but three are often sufficient to prove congruence. Some individually necessary and sufficient conditions for a pair of triangles to be congruent are: In the Euclidean plane, area is defined by comparison with a square of side length 1 {\displaystyle 1} , which has area 1. There are several ways to calculate
10864-591: 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 congruent triangles are also similar, but not all pairs of similar triangles are congruent. Given two congruent triangles, all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have
10976-469: The same term [REDACTED] This disambiguation page lists articles associated with the title Triangle . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Triangle_(disambiguation)&oldid=1242541743 " Categories : Disambiguation pages Place name disambiguation pages Hidden categories: Short description
11088-478: The same way they behave everywhere else in the world." Skeptical researchers , such as Ernest Taves and Barry Singer, have noted how mysteries and the paranormal are very popular and profitable. This has led to the production of vast amounts of material on topics such as the Bermuda Triangle. They were able to show that some of the pro-paranormal material is often misleading or inaccurate, but its producers continue to market it. Accordingly, they have claimed that
11200-413: The sum of any two sides of a triangle is longer than the third side American expression for set square , an object used in engineering and technical drawing, with the aim of providing a straightedge at a right angle or other particular planar angle to a baseline The triangle graph in graph theory Entertainment [ edit ] Music [ edit ] Triangle (musical instrument) , in
11312-403: The teeth of a storm south of Florida on 1 January 1958. Hurricanes are powerful storms which form in tropical waters and have historically cost thousands of lives and caused billions of dollars in damage. The sinking of Francisco de Bobadilla 's Spanish fleet in 1502 was the first recorded instance of a destructive hurricane. These storms have in the past caused a number of incidents related to
11424-488: The three sides and the feet of the three altitudes all lie on a single circle, the triangle's nine-point circle . The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter . The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point ) and the three excircles . The orthocenter (blue point),
11536-491: The triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base b {\displaystyle b} and height h {\displaystyle h} . If two sides a {\displaystyle a} and b {\displaystyle b} and their included angle γ {\displaystyle \gamma } are known, then
11648-402: The triangle is acute; if the circumcenter is located outside the triangle, then the triangle is obtuse. An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side. This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. The length of the altitude
11760-439: The triangle's longest side. Within a given triangle, a longer common side is associated with a smaller inscribed square. If an inscribed square has a side of length q a {\displaystyle q_{a}} and the triangle has a side of length a {\displaystyle a} , part of which side coincides with a side of the square, then q a {\displaystyle q_{a}} ,
11872-1279: The triangle. The Mandart inellipse of a triangle is the ellipse inscribed within the triangle tangent to its sides at the contact points of its excircles. For any ellipse inscribed in a triangle A B C {\displaystyle ABC} , let the foci be P {\displaystyle P} and Q {\displaystyle Q} , then: P A ¯ ⋅ Q A ¯ C A ¯ ⋅ A B ¯ + P B ¯ ⋅ Q B ¯ A B ¯ ⋅ B C ¯ + P C ¯ ⋅ Q C ¯ B C ¯ ⋅ C A ¯ = 1. {\displaystyle {\frac {{\overline {PA}}\cdot {\overline {QA}}}{{\overline {CA}}\cdot {\overline {AB}}}}+{\frac {{\overline {PB}}\cdot {\overline {QB}}}{{\overline {AB}}\cdot {\overline {BC}}}}+{\frac {{\overline {PC}}\cdot {\overline {QC}}}{{\overline {BC}}\cdot {\overline {CA}}}}=1.} From an interior point in
11984-419: The triangles are isosceles whenever they are right pyramids and bipyramids. The Kleetope of a polyhedron is a new polyhedron made by replacing each face of the original with a pyramid, and so the faces of a Kleetope will be triangles. More generally, triangles can be found in higher dimensions, as in the generalized notion of triangles known as the simplex , and the polytopes with triangular facets known as
12096-502: The two aircraft did collide, there were two distinct crash sites, separated by over 160 miles (260 km) of water. However, Kusche's research showed that the unclassified version of the Air Force investigation report revealed that the debris field defining the second "crash site" was examined by a search and rescue ship, and found to be a mass of seaweed and driftwood tangled in an old buoy . The incidents cited above, apart from
12208-1702: The vertices of a triangle, its relative oriented area can be calculated using the shoelace formula , T = 1 2 | x A x B x C y A y B y C 1 1 1 | = 1 2 | x A x B y A y B | + 1 2 | x B x C y B y C | + 1 2 | x C x A y C y A | = 1 2 ( x A y B − x B y A + x B y C − x C y B + x C y A − x A y C ) , {\displaystyle {\begin{aligned}T&={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\y_{A}&y_{B}&y_{C}\\1&1&1\end{vmatrix}}={\tfrac {1}{2}}{\begin{vmatrix}x_{A}&x_{B}\\y_{A}&y_{B}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{B}&x_{C}\\y_{B}&y_{C}\end{vmatrix}}+{\tfrac {1}{2}}{\begin{vmatrix}x_{C}&x_{A}\\y_{C}&y_{A}\end{vmatrix}}\\&={\tfrac {1}{2}}(x_{A}y_{B}-x_{B}y_{A}+x_{B}y_{C}-x_{C}y_{B}+x_{C}y_{A}-x_{A}y_{C}),\end{aligned}}} where | ⋅ | {\displaystyle |\cdot |}
12320-465: The very limits of their range and the slightest error or fault in the equipment could keep them from reaching the small island. On 28 December 1948, a Douglas DC-3 aircraft, number NC16002 , disappeared while on a flight from San Juan, Puerto Rico, to Miami. No trace of the aircraft, or the 32 people on board, was ever found. A Civil Aeronautics Board investigation found there was insufficient information available on which to determine probable cause of
12432-402: Was attributed by Navy investigators to navigational error leading to the aircraft running out of fuel. One of the search and rescue aircraft deployed to look for them, a PBM Mariner with a 13-man crew, also disappeared. A tanker off the coast of Florida reported seeing an explosion and observing a widespread oil slick when fruitlessly searching for survivors. The weather was becoming stormy by
12544-401: Was to blame for the loss. In addition, two of Cyclops ' s sister ships, Proteus and Nereus , were subsequently lost in the North Atlantic during World War II . Both ships were transporting heavy loads of metallic ore similar to that which was loaded on Cyclops during her fatal voyage. In all three cases structural failure due to overloading with a much denser cargo than designed
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