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20,500 | There are approximately 460 globular clusters associated with the Andromeda Galaxy. The most massive of these clusters, identified as Mayall II, nicknamed Globular One, has a greater luminosity than any other known globular cluster in the Local Group of galaxies. It contains several million stars and is about twice as ... | https://en.wikipedia.org/wiki?curid=74331 |
20,501 | Another massive globular cluster, named 037-B327 and discovered in 2006 as is heavily reddened by the Andromeda Galaxy's interstellar dust, was thought to be more massive than G1 and the largest cluster of the Local Group; however, other studies have shown it is actually similar in properties to G1. | https://en.wikipedia.org/wiki?curid=74331 |
20,502 | Unlike the globular clusters of the Milky Way, which show a relatively low age dispersion, Andromeda Galaxy's globular clusters have a much larger range of ages: from systems as old as the galaxy itself to much younger systems, with ages between a few hundred million years to five billion years. | https://en.wikipedia.org/wiki?curid=74331 |
20,503 | In 2005, astronomers discovered a completely new type of star cluster in the Andromeda Galaxy. The new-found clusters contain hundreds of thousands of stars, a similar number of stars that can be found in globular clusters. What distinguishes them from the globular clusters is that they are much larger—several hundred ... | https://en.wikipedia.org/wiki?curid=74331 |
20,504 | The most massive globular cluster in the Andromeda Galaxy, B023-G078, likely has a central intermediate black hole of almost 100,000 solar masses. | https://en.wikipedia.org/wiki?curid=74331 |
20,505 | Like the Milky Way, the Andromeda Galaxy has satellite galaxies, consisting of over 20 known dwarf galaxies. The Andromeda Galaxy's dwarf galaxy population is very similar to the Milky Way's, but the galaxies are much more numerous. The best known and most readily observed satellite galaxies are M32 and M110. Based on ... | https://en.wikipedia.org/wiki?curid=74331 |
20,506 | M110 also appears to be interacting with the Andromeda Galaxy, and astronomers have found in the halo of the latter a stream of metal-rich stars that appear to have been stripped from these satellite galaxies. M110 does contain a dusty lane, which may indicate recent or ongoing star formation. M32 has a young stellar p... | https://en.wikipedia.org/wiki?curid=74331 |
20,507 | The Triangulum Galaxy is a non-dwarf galaxy that lies 750,000 light years from Andromeda. It is currently unknown whether it is a satellite of Andromeda. | https://en.wikipedia.org/wiki?curid=74331 |
20,508 | In 2006, it was discovered that nine of the satellite galaxies lie in a plane that intersects the core of the Andromeda Galaxy; they are not randomly arranged as would be expected from independent interactions. This may indicate a common tidal origin for the satellites. | https://en.wikipedia.org/wiki?curid=74331 |
20,509 | PA-99-N2 was a microlensing event detected in the Andromeda Galaxy in 1999. One of the explanations for this is the gravitational lensing of a red giant by a star with a mass between 0.02 and 3.6 times that of the Sun, which suggested that the star is likely orbited by a planet. This possible exoplanet would have a mas... | https://en.wikipedia.org/wiki?curid=74331 |
20,510 | The Andromeda Galaxy is approaching the Milky Way at about per second. It has been measured approaching relative to the Sun at around as the Sun orbits around the center of the galaxy at a speed of approximately . This makes the Andromeda Galaxy one of about 100 observable blueshifted galaxies. Andromeda Galaxy's tange... | https://en.wikipedia.org/wiki?curid=74331 |
20,511 | Under most viewing conditions, the Andromeda Galaxy is one of the most distant objects that can be seen with the naked eye (M33 and M81 can be seen under very dark skies). The galaxy is commonly located in the sky about the constellations Cassiopeia and Pegasus. Andromeda is best seen during autumn nights in the Northe... | https://en.wikipedia.org/wiki?curid=74331 |
20,512 | The .NET Framework (pronounced as ""dot net"") is a proprietary software framework developed by Microsoft that runs primarily on Microsoft Windows. It was the predominant implementation of the Common Language Infrastructure (CLI) until being superseded by the cross-platform .NET project. It includes a large class libra... | https://en.wikipedia.org/wiki?curid=25143203 |
20,513 | FCL provides the user interface, data access, database connectivity, cryptography, web application development, numeric algorithms, and network communications. Programmers produce software by combining their source code with .NET Framework and other libraries. The framework is intended to be used by most new applicatio... | https://en.wikipedia.org/wiki?curid=25143203 |
20,514 | .NET Framework began as proprietary software, although the firm worked to standardize the software stack almost immediately, even before its first release. Despite the standardization efforts, developers, mainly those in the free and open-source software communities, expressed their unease with the selected terms and t... | https://en.wikipedia.org/wiki?curid=25143203 |
20,515 | In April 2019, Microsoft released .NET Framework 4.8, the last version of the framework as a proprietary offering. Only monthly security and reliability bug fixes to that version have been released since then. No further changes to that version are planned. | https://en.wikipedia.org/wiki?curid=25143203 |
20,516 | Microsoft began developing .NET Framework in the late 1990s, originally under the name of Next Generation Windows Services (NGWS), as part of the .NET strategy. By early 2000, the first beta versions of .NET 1.0 were released. | https://en.wikipedia.org/wiki?curid=25143203 |
20,517 | In August 2000, Microsoft, and Intel worked to standardize Common Language Infrastructure (CLI) and C#. By December 2001, both were ratified Ecma International (ECMA) standards. International Organization for Standardization (ISO) followed in April 2003. The current version of ISO standards are ISO/IEC 23271:2012 and I... | https://en.wikipedia.org/wiki?curid=25143203 |
20,518 | While Microsoft and their partners hold patents for CLI and C#, ECMA and ISO require that all patents essential to implementation be made available under "reasonable and non-discriminatory terms". The firms agreed to meet these terms, and to make the patents available royalty-free. However, this did not apply to the pa... | https://en.wikipedia.org/wiki?curid=25143203 |
20,519 | On October 3, 2007, Microsoft announced that the source code for .NET Framework 3.5 libraries was to become available under the Microsoft Reference Source License (Ms-RSL). The source code repository became available online on January 16, 2008, and included BCL, ASP.NET, ADO.NET, Windows Forms, WPF, and XML. Scott Guth... | https://en.wikipedia.org/wiki?curid=25143203 |
20,520 | The .NET Compact Framework and .NET Micro Framework variants of the .NET Framework provided support for other Microsoft platforms such as Windows Mobile, Windows CE and other resource-constrained embedded devices. Silverlight provided support for web browsers via plug-ins. | https://en.wikipedia.org/wiki?curid=25143203 |
20,521 | In November 2014, Microsoft also produced an update to its patent grants, which further extends the scope beyond its prior pledges. Prior projects like Mono existed in a legal grey area because Microsoft's earlier grants applied only to the technology in "covered specifications", including strictly the 4th editions eac... | https://en.wikipedia.org/wiki?curid=25143203 |
20,522 | On March 31, 2016, Microsoft announced at Microsoft Build that they will completely relicense Mono under an MIT License even in scenarios where formerly a commercial license was needed. Microsoft also supplemented its prior patent promise for Mono, stating that they will not assert any "applicable patents" against part... | https://en.wikipedia.org/wiki?curid=25143203 |
20,523 | Microsoft's press release highlights that the cross-platform commitment now allows for a fully open-source, modern server-side .NET stack. Microsoft released the source code for WPF, Windows Forms and WinUI on December 4, 2018. | https://en.wikipedia.org/wiki?curid=25143203 |
20,524 | Common Language Infrastructure (CLI) provides a language-neutral platform for application development and execution. By implementing the core aspects of .NET Framework within the scope of CLI, these functions will not be tied to one language but will be available across the many languages supported by the framework. | https://en.wikipedia.org/wiki?curid=25143203 |
20,525 | .NET Framework includes the Common Language Runtime (CLR). It serves as the execution engine of .NET Framework and offers many services such as memory management, type safety, exception handling, garbage collection, security and thread management. All programs written for .NET Framework are executed by the CLR. | https://en.wikipedia.org/wiki?curid=25143203 |
20,526 | Programs written for .NET Framework are compiled into Common Intermediate Language code (CIL), as opposed to being directly compiled into machine code. During execution, an architecture-specific just-in-time compiler (JIT) turns the CIL code into machine code. | https://en.wikipedia.org/wiki?curid=25143203 |
20,527 | Compiled CIL code is stored in CLI assemblies. As mandated by the specification, assemblies are stored in Portable Executable (PE) file format, common on Windows platform for all "dynamic-link library" (DLL) and "executable" EXE files. Each assembly consists of one or more files, one of which must contain a manifest be... | https://en.wikipedia.org/wiki?curid=25143203 |
20,528 | A private key can also be used by the creator of the assembly for strong naming. The public key token identifies which private key an assembly is signed with. Only the creator of the key pair (typically the person signing the assembly) can sign assemblies that have the same strong name as a prior version assembly, sinc... | https://en.wikipedia.org/wiki?curid=25143203 |
20,529 | Starting with Visual Studio 2015, .NET Native compilation technology allows for the compilation of .NET code of Universal Windows Platform apps directly to machine code rather than CIL code, but the app must be written in either C# or Visual Basic.NET. | https://en.wikipedia.org/wiki?curid=25143203 |
20,530 | .NET Framework includes an implementation of the CLI foundational Standard Libraries. The .NET Framework Class Library (FCL) is organized in a hierarchy of namespaces. Most of the built-in application programming interfaces (APIs) are part of either codice_1 or codice_2 namespaces. These class libraries implement many ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,531 | BCL includes a small subset of the entire class library and is the core set of classes that serve as the basic API of CLR. For .NET Framework most classes considered being part of BCL reside in codice_3, codice_4 and codice_5. BCL classes are available in .NET Framework as well as its alternative implementations includ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,532 | FCL refers to the entire class library that ships with .NET Framework. It includes an expanded set of libraries, including BCL, Windows Forms, ASP.NET, and Windows Presentation Foundation (WPF) but also extensions to the base class libraries ADO.NET, Language Integrated Query (LINQ), Windows Communication Foundation (W... | https://en.wikipedia.org/wiki?curid=25143203 |
20,533 | With the introduction of alternative implementations (e.g., Silverlight), Microsoft introduced the concept of Portable Class Libraries (PCL) allowing a consuming library to run on more than one platform. With the further proliferation of .NET platforms, the PCL approach failed to scale (PCLs are defined intersections o... | https://en.wikipedia.org/wiki?curid=25143203 |
20,534 | NuGet is the package manager for all .NET platforms. It is used to retrieve third-party libraries into a .NET project with a global library feed at NuGet.org. Private feeds can be maintained separately, e.g., by a build server or a file system directory. | https://en.wikipedia.org/wiki?curid=25143203 |
20,535 | Microsoft introduced C++/CLI in Visual Studio 2005, which is a language and means of compiling Visual C++ programs to run within the .NET Framework. Some parts of the C++ program still run within an unmanaged Visual C++ Runtime, while specially modified parts are translated into CIL code and run with the .NET Framework... | https://en.wikipedia.org/wiki?curid=25143203 |
20,536 | Assemblies compiled using the C++/CLI compiler are termed mixed-mode assemblies since they contain native and managed code in the same DLL. Such assemblies are more complex to reverse engineer since .NET decompilers such as .NET Reflector reveal only the managed code. | https://en.wikipedia.org/wiki?curid=25143203 |
20,537 | Because computer systems commonly require interaction between newer and older applications, .NET Framework provides means to access functions implemented in newer and older programs that execute outside .NET environment. Access to Component Object Model (COM) components is provided in codice_7 and codice_8 namespaces o... | https://en.wikipedia.org/wiki?curid=25143203 |
20,538 | Interoperability enables you to preserve and take advantage of existing investments in unmanaged code. Code that runs under the control of the common language runtime (CLR) is called managed code, and code that runs outside the CLR is called unmanaged code | https://en.wikipedia.org/wiki?curid=25143203 |
20,539 | .NET Framework introduces a Common Type System (CTS) that defines all possible data types and programming constructs supported by CLR and how they may or may not interact conforming to CLI specifications. Because of this feature, .NET Framework supports the exchange of types and object instances between libraries and a... | https://en.wikipedia.org/wiki?curid=25143203 |
20,540 | CTS and the CLR used in .NET Framework also enforce type safety. This prevents ill-defined casts, wrong method invocations, and memory size issues when accessing an object. This also makes most CLI languages statically typed (with or without type inference). However, starting with .NET Framework 4.0, the Dynamic Langua... | https://en.wikipedia.org/wiki?curid=25143203 |
20,541 | While Microsoft has never implemented the full framework on any system except Microsoft Windows, it has engineered the framework to be cross-platform, and implementations are available for other operating systems (see Silverlight and § Alternative implementations). Microsoft submitted the specifications for CLI (which ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,542 | .NET Framework has its own security mechanism with two general features: Code Access Security (CAS), and validation and verification. CAS is based on evidence that is associated with a specific assembly. Typically the evidence is the source of the assembly (whether it is installed on the local machine or has been downl... | https://en.wikipedia.org/wiki?curid=25143203 |
20,543 | Managed CIL bytecode is easier to reverse-engineer than native code, unless obfuscated. decompiler programs enable developers with no reverse-engineering skills to view the source code behind unobfuscated .NET assemblies. In contrast, apps compiled to native machine code are much harder to reverse-engineer, and source ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,544 | CLR frees the developer from the burden of managing memory (allocating and freeing up when done); it handles memory management itself by detecting when memory can be safely freed. Instantiations of .NET types (objects) are allocated from the managed heap; a pool of memory managed by CLR. As long as a reference to an ob... | https://en.wikipedia.org/wiki?curid=25143203 |
20,545 | .NET Framework includes a garbage collector (GC) which runs periodically, on a separate thread from the application's thread, that enumerates all the unusable objects and reclaims the memory allocated to them. It is a non-deterministic, compacting, mark-and-sweep garbage collector. GC runs only when a set amount of mem... | https://en.wikipedia.org/wiki?curid=25143203 |
20,546 | The garbage collector used by .NET Framework is also "generational". Objects are assigned a "generation". Newly created objects are tagged "Generation 0". Objects that survive one garbage collection are tagged "Generation 1". Generation 1 objects that survive another collection are "Generation 2". The framework uses up... | https://en.wikipedia.org/wiki?curid=25143203 |
20,547 | When an application is first launched, the .NET Framework compiles the CIL code into executable code using its just-in-time compiler, and caches the executable program into the .NET Native Image Cache. Due to caching, the application launches faster for subsequent launches, although the first launch is usually slower. ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,548 | The garbage collector, which is integrated into the environment, can introduce unanticipated delays of execution over which the developer has little direct control. "In large applications, the number of objects that the garbage collector needs to work with can become very large, which means it can take a very long time... | https://en.wikipedia.org/wiki?curid=25143203 |
20,549 | .NET Framework provides support for calling Streaming SIMD Extensions (SSE) via managed code from April 2014 in Visual Studio 2013 Update 2. However, Mono has provided support for SIMD Extensions as of version 2.2 within the namespace in 2009. Mono's lead developer Miguel de Icaza has expressed hope that this SIMD supp... | https://en.wikipedia.org/wiki?curid=25143203 |
20,550 | .NET Framework was the predominant implementation of .NET technologies, until the release of .NET. Other implementations for parts of the framework exist. Although the runtime engine is described by an ECMA-ISO specification, other implementations of it may be encumbered by patent issues; ISO standards may include the ... | https://en.wikipedia.org/wiki?curid=25143203 |
20,551 | In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other ... | https://en.wikipedia.org/wiki?curid=26513034 |
20,552 | The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. | https://en.wikipedia.org/wiki?curid=26513034 |
20,553 | When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. | https://en.wikipedia.org/wiki?curid=26513034 |
20,554 | The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but "n"-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical... | https://en.wikipedia.org/wiki?curid=26513034 |
20,555 | If "c" denotes the length of the hypotenuse and "a" and "b" denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: | https://en.wikipedia.org/wiki?curid=26513034 |
20,556 | If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation | https://en.wikipedia.org/wiki?curid=26513034 |
20,557 | If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as | https://en.wikipedia.org/wiki?curid=26513034 |
20,558 | A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation. | https://en.wikipedia.org/wiki?curid=26513034 |
20,559 | In one rearrangement proof, two squares are used whose sides have a measure of formula_6 and which contain four right triangles whose sides are a, b and c, with the hypotenuse being c. In the square on the right side, the triangles are placed such that the corners of the square correspond to the corners of the right an... | https://en.wikipedia.org/wiki?curid=26513034 |
20,560 | In another proof rectangles in the second box can also be placed such that both have one corner that correspond to consecutive corners of the square. In this way they also form two boxes, this time in consecutive corners, with areas formula_16 and formula_17which will again lead to a second square of with the area form... | https://en.wikipedia.org/wiki?curid=26513034 |
20,561 | English mathematician Sir Thomas Heath gives this proof in his commentary on Proposition I.47 in Euclid's "Elements", and mentions the proposals of German mathematicians Carl Anton Bretschneider and Hermann Hankel that Pythagoras may have known this proof. Heath himself favors a different proposal for a Pythagorean pro... | https://en.wikipedia.org/wiki?curid=26513034 |
20,562 | The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side "c", as shown in the lower part of the diagram. This results in a larger square, with side and area . The four triangles and the square side "c" must have the same area as the larger square, | https://en.wikipedia.org/wiki?curid=26513034 |
20,563 | A similar proof uses four copies of a right triangle with sides "a", "b" and "c", arranged inside a square with side "c" as in the top half of the diagram. The triangles are similar with area formula_21, while the small square has side and area . The area of the large square is therefore | https://en.wikipedia.org/wiki?curid=26513034 |
20,564 | This theorem may have more known proofs than any other (the law of quadratic reciprocity being another contender for that distinction); the book "The Pythagorean Proposition" contains 370 proofs. | https://en.wikipedia.org/wiki?curid=26513034 |
20,565 | This proof is based on the proportionality of the sides of three similar triangles, that is, upon the fact that the ratio of any two corresponding sides of similar triangles is the same regardless of the size of the triangles. | https://en.wikipedia.org/wiki?curid=26513034 |
20,566 | Let "ABC" represent a right triangle, with the right angle located at "C", as shown on the figure. Draw the altitude from point "C", and call "H" its intersection with the side "AB". Point "H" divides the length of the hypotenuse "c" into parts "d" and "e". The new triangle, "ACH," is similar to triangle "ABC", because... | https://en.wikipedia.org/wiki?curid=26513034 |
20,567 | The first result equates the cosines of the angles "θ", whereas the second result equates their sines. | https://en.wikipedia.org/wiki?curid=26513034 |
20,568 | The role of this proof in history is the subject of much speculation. The underlying question is why Euclid did not use this proof, but invented another. One conjecture is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the "Elements", and that the theory of pr... | https://en.wikipedia.org/wiki?curid=26513034 |
20,569 | In outline, here is how the proof in Euclid's "Elements" proceeds. The large square is divided into a left and right rectangle. A triangle is constructed that has half the area of the left rectangle. Then another triangle is constructed that has half the area of the square on the left-most side. These two triangles are... | https://en.wikipedia.org/wiki?curid=26513034 |
20,570 | Let "A", "B", "C" be the vertices of a right triangle, with a right angle at "A". Drop a perpendicular from "A" to the side opposite the hypotenuse in the square on the hypotenuse. That line divides the square on the hypotenuse into two rectangles, each having the same area as one of the two squares on the legs. | https://en.wikipedia.org/wiki?curid=26513034 |
20,571 | Next, each top square is related to a triangle congruent with another triangle related in turn to one of two rectangles making up the lower square. | https://en.wikipedia.org/wiki?curid=26513034 |
20,572 | This proof, which appears in Euclid's "Elements" as that of Proposition 47 in Book 1, demonstrates that the area of the square on the hypotenuse is the sum of the areas of the other two squares. | https://en.wikipedia.org/wiki?curid=26513034 |
20,573 | This is quite distinct from the proof by similarity of triangles, which is conjectured to be the proof that Pythagoras used. | https://en.wikipedia.org/wiki?curid=26513034 |
20,574 | Another by rearrangement is given by the middle animation. A large square is formed with area "c", from four identical right triangles with sides "a", "b" and "c", fitted around a small central square. Then two rectangles are formed with sides "a" and "b" by moving the triangles. Combining the smaller square with these... | https://en.wikipedia.org/wiki?curid=26513034 |
20,575 | The third, rightmost image also gives a proof. The upper two squares are divided as shown by the blue and green shading, into pieces that when rearranged can be made to fit in the lower square on the hypotenuse – or conversely the large square can be divided as shown into pieces that fill the other two. This way of cut... | https://en.wikipedia.org/wiki?curid=26513034 |
20,576 | Albert Einstein gave a proof by dissection in which the pieces do not need to be moved. Instead of using a square on the hypotenuse and two squares on the legs, one can use any other shape that includes the hypotenuse, and two similar shapes that each include one of two legs instead of the hypotenuse (see Similar figur... | https://en.wikipedia.org/wiki?curid=26513034 |
20,577 | As shown in the accompanying animation, area-preserving shear mappings and translations can transform the squares on the sides adjacent to the right-angle onto the square on the hypotenuse, together covering it exactly. Each shear leaves the base and height unchanged, thus leaving the area unchanged too. The translatio... | https://en.wikipedia.org/wiki?curid=26513034 |
20,578 | A related proof was published by future U.S. President James A. Garfield (then a U.S. Representative) (see diagram). Instead of a square it uses a trapezoid, which can be constructed from the square in the second of the above proofs by bisecting along a diagonal of the inner square, to give the trapezoid as shown in th... | https://en.wikipedia.org/wiki?curid=26513034 |
20,579 | The inner square is similarly halved, and there are only two triangles so the proof proceeds as above except for a factor of formula_29, which is removed by multiplying by two to give the result. | https://en.wikipedia.org/wiki?curid=26513034 |
20,580 | One can arrive at the Pythagorean theorem by studying how changes in a side produce a change in the hypotenuse and employing calculus. | https://en.wikipedia.org/wiki?curid=26513034 |
20,581 | The triangle "ABC" is a right triangle, as shown in the upper part of the diagram, with "BC" the hypotenuse. At the same time the triangle lengths are measured as shown, with the hypotenuse of length "y", the side "AC" of length "x" and the side "AB" of length "a", as seen in the lower diagram part. | https://en.wikipedia.org/wiki?curid=26513034 |
20,582 | If "x" is increased by a small amount "dx" by extending the side "AC" slightly to "D", then "y" also increases by "dy". These form two sides of a triangle, "CDE", which (with "E" chosen so "CE" is perpendicular to the hypotenuse) is a right triangle approximately similar to "ABC". Therefore, the ratios of their sides m... | https://en.wikipedia.org/wiki?curid=26513034 |
20,583 | This can be rewritten as formula_31 , which is a differential equation that can be solved by direct integration: | https://en.wikipedia.org/wiki?curid=26513034 |
20,584 | This is more of an intuitive proof than a formal one: it can be made more rigorous if proper limits are used in place of "dx" and "dy". | https://en.wikipedia.org/wiki?curid=26513034 |
20,585 | Given a triangle with sides of length "a", "b", and "c", if then the angle between sides "a" and "b" is a right angle. | https://en.wikipedia.org/wiki?curid=26513034 |
20,586 | For any three positive real numbers "a", "b", and "c" such that , there exists a triangle with sides "a", "b" and "c" as a consequence of the converse of the triangle inequality. | https://en.wikipedia.org/wiki?curid=26513034 |
20,587 | This converse appears in Euclid's "Elements" (Book I, Proposition 48): "If in a triangle the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right." | https://en.wikipedia.org/wiki?curid=26513034 |
20,588 | Let "ABC" be a triangle with side lengths "a", "b", and "c", with Construct a second triangle with sides of length "a" and "b" containing a right angle. By the Pythagorean theorem, it follows that the hypotenuse of this triangle has length "c" = , the same as the hypotenuse of the first triangle. Since both triangles' ... | https://en.wikipedia.org/wiki?curid=26513034 |
20,589 | The above proof of the converse makes use of the Pythagorean theorem itself. The converse can also be proven without assuming the Pythagorean theorem. | https://en.wikipedia.org/wiki?curid=26513034 |
20,590 | A corollary of the Pythagorean theorem's converse is a simple means of determining whether a triangle is right, obtuse, or acute, as follows. Let "c" be chosen to be the longest of the three sides and (otherwise there is no triangle according to the triangle inequality). The following statements apply: | https://en.wikipedia.org/wiki?curid=26513034 |
20,591 | Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: | https://en.wikipedia.org/wiki?curid=26513034 |
20,592 | where "α" is the angle opposite to side "a", "β" is the angle opposite to side "b", "γ" is the angle opposite to side "c", and sgn is the sign function. | https://en.wikipedia.org/wiki?curid=26513034 |
20,593 | A Pythagorean triple has three positive integers "a", "b", and "c", such that In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Such a triple is commonly written Some well-known examples are and | https://en.wikipedia.org/wiki?curid=26513034 |
20,594 | A primitive Pythagorean triple is one in which "a", "b" and "c" are coprime (the greatest common divisor of "a", "b" and "c" is 1). | https://en.wikipedia.org/wiki?curid=26513034 |
20,595 | Given a right triangle with sides formula_35 and altitude formula_36 (a line from the right angle and perpendicular to the hypotenuse formula_37). The Pythagorean theorem has, | https://en.wikipedia.org/wiki?curid=26513034 |
20,596 | where formula_43 for any non-zero real formula_44. If the formula_45 are to be integers, the smallest solution formula_46 is then | https://en.wikipedia.org/wiki?curid=26513034 |
20,597 | using the smallest Pythagorean triple formula_48. The reciprocal Pythagorean theorem is a special case of the optic equation | https://en.wikipedia.org/wiki?curid=26513034 |
20,598 | where the denominators are squares and also for a heptagonal triangle whose sides formula_50 are square numbers. | https://en.wikipedia.org/wiki?curid=26513034 |
20,599 | One of the consequences of the Pythagorean theorem is that line segments whose lengths are incommensurable (so the ratio of which is not a rational number) can be constructed using a straightedge and compass. Pythagoras' theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is rel... | https://en.wikipedia.org/wiki?curid=26513034 |
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