① <book> How much value ~? Your time has been ~. ② short geometry.
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jǐ hé
: The number (for ask)
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No. 3
Carry on in geometry. - "Warring Zhao policy"
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No. 4
Luo Fu-year geometry. - "Folk Song and Ballad Mo Shang Sang"
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No. 5
Killed geometry. - Tang Li Zhaowei "Liuyi"
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No. 6
Quite different to geometry. - Ming Liu Ji "sincerity Bo Wen-Cheng Liu of public Collection"
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No. 7
Worth.
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No. 8
: Geometry referred to as
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No. 9
Still a number of how many. "Poetry Xiaoya fair speech": "To still be more Ulgu only geometric?" Ma Ruichen and Interpretation: "Seoul Home only geometry, that is, only geometry is also made Seoul." "Historical Records of white from WANG Jian Biography": " So the First Emperor Shun Lee asked: 'Jing Wu Yu capture, in general a person who used a geometric enough?' "" New Tang Lee Da Zuo Chuan ":" ﹝ ﹞ Chang Cambodia is the calm that said: 'General Home North gate geometry? 'said:' Three years carry on '. "Qing Liu Xianting" Guang Yang Miscellaneous notes, "volume IV:" Kid fee is not cheap now! furniture geometry, is that Hu is Lord! "" Travels "Third Round: "High public asked: 'drug money to consult geometry?'" Guo Xiaochuan "Spring Song" Two: "Battle of poetry can hold 1000 million baskets baskets, and my words do, but also geometry!"
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No. 10
Mathematics in a Division. See "Geometry."
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No. 11
Geometry (jǐ-): ① number: its achievements have geometry | I do not know the cost yet to geometry? ② Mathematics Division 1. See "Geometry."
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No. 12
A number (for ask) Carry on geometry. - "Warring Zhao policy" Luo Fu in geometry. - "Folk Song and Ballad Mo Mulberry" Killed geometry. - Tang Li Zhaowei "Liu Yi Chuan" Quite different to geometry. - Ming Liu Ji "sincerity of public Collection Bo Liu Wencheng" Worth. "Geometric" origin of the name - scientists Xu Scroll down scroll up and down In 1594, Xu in Shaozhou (now Guangdong Shaoguan) teaching time, met a spread of Catholicism in China Guo Jesuit home soil. Guo living in there, he first saw a map of the world outside of China as much as you know in a big world; they first heard the earth is round, a man named Magellan Westerners boat trip around the earth ring week; also Italian scientist Galileo first heard of the astronomical telescope manufacturer, can clearly observed the operation of the stars of heaven. All of these, for him, are unheard of novelty. Since then, he came into contact with modern Western science, knowledge has been enriched. Ming Dynasty, eunuch, political darkness, the people's life is very painful, full of peasant uprisings occurred; is on the rise of the Manchu nobility Northeast and do not attack when the Ming Dynasty, the whole society in a turbulent state. Like all honest intellectuals, Xu full of patriotic enthusiasm, he wanted to use technology to help the country prosperous and strong, so that Lebanon had on the world's "rich food, clothing, never hungry," the stability and prosperous life. He therefore concluded that not only should be seriously scientific achievements in ancient China, the West should also be good to learn the advanced natural sciences, each other, so our further development of science and technology. Exchanges in the same home when Guo Jing, Xu told China to missionary Matteo Ricci, the Jesuit president of Western natural science proficiency, to inquire about his whereabouts around, want to be facing him for advice. In 1600, he received the news of missionary Matteo Ricci in Nanjing, which made a special trip to Nanjing to visit. Ricci is an Italian, the original Ming Jiaoma too Olic. He always studious, mathematics, physics, astronomy, medicine is very accomplished, and good at making watches and clocks, sundial (gui gui, sundial is an ancient instrument measuring time), good at drawing maps and sculpture. Three-year-old graduated from the Theological College, the Jesuit Matteo Ricci was sent to China to preach. In order to facilitate his contacts with the Chinese people, hard to learn Chinese language, writing and ancient culture, and put China's clothing, according to Chinese etiquette and customs activities, but also for their own took a Chinese name Ricci. After a period of study, Xu fully understand the contents of this book of Euclid, deeply for its basic theory and logical reasoning are impressed that these are precisely of the inadequacy of ancient mathematics. He felt that although China's ancient mathematics has made glorious achievements, but for thousands of years of experience has been limited evidence, not a good use of logical reasoning methods. This book if it can be introduced over Euclid, development of mathematics in China will be very beneficial. Thus, Xu suggested Ricci to cooperate with him, along with it into Chinese. Start, Ricci quite hesitant about this proposal, because this book is Euclid's written in Latin, Latin and Chinese grammar is different from very different vocabulary, many mathematics books in the Chinese terms there are no corresponding ready-made vocabulary. Translation may be accurate, fluent but understandable, is not easy. Earlier there was a family name of Chiang lifts off with the translation of Matteo Ricci co-pilot, because this reason had to give up halfway. But Xu is very confident that as long as he is willing to work hard to rack their brains, careful scrutiny, repeated changes can always be translated into the. In his repeated persuasion, Ricci permission. From the busy winter snow peach and plum blossoms in the spring of next year, Xu and Ricci translated the first six volumes of this work. Xu would like to bang, and then down the translation, for translation in the year after nine volumes, but Ricci has advocated engraving first published in six volumes before and listen to reflect say. Before printing, Xu and the translation process alone, polished three times as much as possible to change the translation was accurate. Then he together with Matteo Ricci, the translation of the title common to finalize. This book of the Latin original called "Euclidean original", if Direct translated into Chinese, not the elephant is a math book. If in accordance with its contents, into "the original form of learning", and it looks very dated. Ricci said that the Chinese in the "form school", the English called "geo", it is intended to measure the meaning of the Greek land, can find in the Chinese vocabulary with which it has a similar pronunciation words of similar meaning are also . Xu traced a dozen phrases, are not ideal. Then he remembered the "Geometry" word, that it "geo" meaning cut the sounds, suggested the title translated as "Geometry", Ricci was very satisfied. 1607, "Geometry" was published six volumes before and immediately caused a huge reaction, has become engaged in mathematical work of the Ming and Qing Dynasty's a must read for the development of our country played an important role in modern mathematics. Later, Xu and although no longer able to Matteo Ricci translated with "Elements" of the back nine volumes, but gradually he wrote many other scientific works, especially the "agricultural policy book," this monumental work, in our country and the world the history of science has an important position. Future generations of people, to commemorate Xu's outstanding contributions in science, France and China put his hometown was renamed Department of Xujiahui. Ancient geometry == == Period of Chinese civilization and its corresponding level of development of civilization rather, it may also have the same advanced math, but no traces of that time allows us to confirm this. Perhaps this is partly because of China's early use of the original paper, instead of clay or stone to record their achievements. == == Origin of the name The word geometry comes from the earliest Greek "γεωμετρία", the "γέα" (land) and "μετρε ĭν" (measurement) synthesized from two words, means the land survey, which measured to surgery. Later Latin into "geometria". Chinese in the "Geometry" word, the first is Matteo Ricci in the Ming Dynasty, Xu joint translation "Geometry", by Xu created by. By that time did not give the basis for much later that the hand geometry may be the Latin transliteration of the Greek geo, on the other hand the "Geometry" is also a way of using geometry to illustrate the content of number theory, it could be magnitude (number of ) of the translation, it is generally believed that the geometry geometria sound, meaning and translation. Published in 1607, "Geometry" in the translation on the geometry at the time did not pass, there is also another contemporary translation - shaped, for instance by Mateer, Zou Liwen, Liu Yongxi compiled "form of learning preparedness purpose" at that time also a certain influence. In 1857, Li Shan-Ian, Alexander Wylie continued translation of "Geometry" was published after the 9 volumes, although the name of geometry has been some attention, but until the early 20th century, when only the more obvious word to replace the morphological trends, such as the 1910 "aims to shape school equipment" turned the 11th printing in Chengdu will be renamed the printed edition Xu Shuxun "added geometry." Until the mid-20th century, has little "form school" time use appear. == == Branch Plane geometry Solid Geometry Non-Euclidean geometry Roche geometry Riemannian geometry Analytic Geometry Projective geometry Affine geometry Algebraic Geometry Differential Geometry Computational Geometry Topology Fractal geometry Expand knowledge】 【 Three ancient Greek geometric drawing question is: ① square the circle, seeking to make a square, an area equal to a given circle; ② trisection of any angle; ③ times the cube, seeking to make a cube, its volume is known to double the cube. The difficulty of these problems is the mapping can only ruler (without scale, only a linear scale) and compasses. After two thousand years of exploration, and finally prove that the restrictions in the ruler, it is simply impossible to make the required graphics. Circle and square are common graphics, how to make a square with a ruler known as a round of product? Historically, perhaps no geometry problems like this "square the circle" problem that aroused strong interest. As early as the 5th century BC, there are many people studying the issue, the Greeks used for such activities, a special word "" to mean, which means "dedicated to square the circle problem", we can see things are quite common. This problem is the first researcher Anna Sago Douglas, he was "ungodly" to the charges in jail, in prison, square the circle of painstaking research problem. Researchers have known since Hippocrates, Antioch Feng, Hippy Elias et al. Antioch Fung proposed a "method of exhaustion", is the prototype of modern limit theory. First be inscribed square (or is 6 gon), and each time doubling the number of edges, have internal 8,16,32, ... gon, he believes that "last" regular polygon will coincide with the circumference. This will square the circle was. Conclusion is wrong, but it has provided a method for seeking the approximate area of a circle, Archimedes calculated pi to become the leading method. Liu Hui of China cutting circle method coincide. 5th century BC, Athens "Homo sapiens School" to the above questions centered research. Because of the ruler can not be used to solve, and often people go into new areas. For example inspired conic, circular curve, and cut three or four times the number found in algebraic curve. Since the establishment of the 17th century analytic geometry, the possibility of ruler and compasses only guidelines. Wang Wenzel 1837 pl give any angle trisection can not be used and times of cubic proof Ruler 1882 clfvon Lindemann proved the transcendence of π, square the circle can be established without possibility of . 1895 (c.) f. Klein summarizes the previous research, the "three geometric problems" (Chinese translation, 1930) a book, three major issues can not be given to mapping using simple ruler card method, solve the outstanding issues over two thousand years. . ∠ cob = 1 / 3 ∠ acb The tools used here have been limited to the ruler, and the mapping methods with the Public sub. The other two problems can be resolved by other tools.
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A number (for ask)
Carry on geometry. - "Warring Zhao policy" Luo Fu in geometry. - "Folk Song and Ballad Mo Mulberry" Killed geometry. - Tang Li Zhaowei "Liu Yi Chuan" Quite different to geometry. - Ming Liu Ji "sincerity of public Collection Bo Liu Wencheng" Worth.
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"Geometric" origin of the name - scientists Xu
In 1594, Xu in Shaozhou (now Guangdong Shaoguan) teaching time, met a spread of Catholicism in China Guo Jesuit home soil. Guo living in there, he first saw a map of the world outside of China as much as you know in a big world; they first heard the earth is round, a man named Magellan Westerners boat trip around the earth ring week; also Italian scientist Galileo first heard of the astronomical telescope manufacturer, can clearly observed the operation of the stars of heaven. All of these, for him, are unheard of novelty. Since then, he came into contact with modern Western science, knowledge has been enriched. Ming Dynasty, eunuch, political darkness, the people's life is very painful, full of peasant uprisings occurred; is on the rise of the Manchu nobility Northeast and do not attack when the Ming Dynasty, the whole society in a turbulent state. Like all honest intellectuals, Xu full of patriotic enthusiasm, he wanted to use technology to help the country prosperous and strong, so that Lebanon had on the world's "rich food, clothing, never hungry," the stability and prosperous life. He therefore concluded that not only should be seriously scientific achievements in ancient China, the West should also be good to learn the advanced natural sciences, each other, so our further development of science and technology. Exchanges in the same home when Guo Jing, Xu told China to missionary Matteo Ricci, the Jesuit president of Western natural science proficiency, to inquire about his whereabouts around, want to be facing him for advice. In 1600, he received the news of missionary Matteo Ricci in Nanjing, which made a special trip to Nanjing to visit. Is the Italian Matteo Ricci, the former Ming Jiaoma too Olic. He always studious, mathematics, physics, astronomy, medicine is very accomplished, and good at making watches and clocks, sundial (guĭ, sundial is an ancient instrument measuring time), good at drawing maps and sculpture. Three-year-old graduated from the Theological College, the Jesuit Matteo Ricci was sent to China to preach. In order to facilitate his contacts with the Chinese people, hard to learn Chinese language, writing and ancient culture, and put China's clothing, according to Chinese etiquette and customs activities, but also for their own took a Chinese name Ricci. After a period of study, Xu fully understand the contents of this book of Euclid, deeply for its basic theory and logical reasoning are impressed that these are precisely of the inadequacy of ancient mathematics. He felt that although China's ancient mathematics has made glorious achievements, but for thousands of years of experience has been limited evidence, not a good use of logical reasoning methods. This book if it can be introduced over Euclid, development of mathematics in China will be very beneficial. Thus, Xu suggested Ricci to cooperate with him, along with it into Chinese. Start, Ricci quite hesitant about this proposal, because this book is Euclid's written in Latin, Latin and Chinese grammar is different from very different vocabulary, many mathematics books in the Chinese terms there are no corresponding ready-made vocabulary. Translation may be accurate, fluent but understandable, is not easy. Earlier there was a family name of Chiang lifts off with the translation of Matteo Ricci co-pilot, because this reason had to give up halfway. But Xu is very confident that as long as he is willing to work hard to rack their brains, careful scrutiny, repeated changes can always be translated into the. In his repeated persuasion, Ricci permission. From the busy winter snow peach and plum blossoms in the spring of next year, Xu and Ricci translated the first six volumes of this work. Xu would like to bang, and then down the translation, for translation in the year after nine volumes, but Ricci has advocated engraving first published in six volumes before and listen to reflect say. Before printing, Xu and the translation process alone, polished three times as much as possible to change the translation was accurate. Then he together with Matteo Ricci, the translation of the title common to finalize. This book of the Latin original called "Euclidean original", if Direct translated into Chinese, not the elephant is a math book. If in accordance with its contents, into "the original form of learning", and it looks very dated. Ricci said that the Chinese in the "form school", the English called "Geo", it is intended to measure the meaning of the Greek land, can find in the Chinese vocabulary with which it has a similar pronunciation words of similar meaning are also . Xu traced a dozen phrases, are not ideal. Then he remembered the "Geometry" word, that it "Geo" sounds almost Italian cut, suggested the title translated as "Geometry", Ricci was very satisfied. 1607, "Geometry" was published six volumes before and immediately caused a huge reaction, has become engaged in mathematical work of the Ming and Qing Dynasty's a must read for the development of our country played an important role in modern mathematics. Later, Xu and although no longer able to Matteo Ricci translated with "Elements" of the back nine volumes, but gradually he wrote many other scientific works, especially the "agricultural policy book," this monumental work, in our country and the world the history of science has an important position. Future generations of people, to commemorate Xu's outstanding contributions in science, France and China put his hometown was renamed Department of Xujiahui.
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Ancient geometry == ==
Period of Chinese civilization and its corresponding level of development of civilization rather, it may also have the same advanced math, but no traces of that time allows us to confirm this. Perhaps this is partly because of China's early use of the original paper, instead of clay or stone to record their achievements.
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== == Origin of the name
The word geometry comes from the earliest Greek "γεωμετρία", the "γέα" (land) and "μετρε ĭν" (measurement) synthesized from two words, means the land survey, which measured to surgery. Later Latin into "geometria". Chinese in the "Geometry" word, the first is Matteo Ricci in the Ming Dynasty, Xu joint translation "Geometry", by Xu created by. By that time did not give the basis for much later that the hand geometry may be the Latin transliteration of the Greek GEO other hand, the "geometry" also has described using the geometric approach to the content of number theory, it could be magnitude (number of ) of the translation, it is generally believed that the geometry geometria sound, meaning and translation. Published in 1607, "Geometry" in the translation on the geometry at the time did not pass, there is also another contemporary translation - shaped, for instance by Mateer, Zou Liwen, Liu Yongxi compiled "form of learning preparedness purpose" at that time also a certain influence. In 1857, Li Shan-Ian, Alexander Wylie continued translation of "Geometry" was published after the 9 volumes, although the name of geometry has been some attention, but until the early 20th century, when only the more obvious word to replace the morphological trends, such as the 1910 "aims to shape school preparation," printed in Chengdu 11th printed edition Xu Shuxun turn will be renamed the "continued geometry." Until the mid-20th century, has little "form school" use of the term appears.
Three ancient Greek geometric drawing question is: ① square the circle, seeking to make a square, an area equal to a given circle; ② trisection of any angle; ③ times the cube, seeking to make a cube, its volume is known to double the cube. The difficulty of these problems is the mapping can only ruler (without scale, only a linear scale) and compasses. After two thousand years of exploration, and finally prove that the restrictions in the ruler, it is simply impossible to make the required graphics. Circle and square are common graphics, how to make a square with a ruler known as a round of product? Historically, perhaps no geometry problems like this "square the circle" problem that aroused strong interest. As early as the 5th century BC, there are many people studying the issue, the Greeks used for such activities, a special word "" to mean, which means "dedicated to square the circle problem", we can see things are quite common. This problem is the first researcher Anna Sago Douglas, he was "ungodly" to the charges in jail, in prison, square the circle of painstaking research problem. Researchers have known since Hippocrates, Antioch Feng, Hippy Elias et al. Antioch Fung proposed a "method of exhaustion", is the prototype of modern limit theory. First be inscribed square (or is 6 gon), and each time doubling the number of edges, have internal 8,16,32, ... gon, he believes that "last" regular polygon will coincide with the circumference. This will square the circle was. Conclusion is wrong, but it has provided a method for seeking the approximate area of a circle, Archimedes calculated pi to become the leading method. Liu Hui of China cutting circle method coincide. Bisection with an angle ruler is easy, for some angle, such as 90 °, 135 °, 180 °, trisection is not difficult. Naturally raised any angle trisection problem. Such as the 60 ° angle trisection able to connect to regular 18-gon and is 9-gon, angle trisection problem is caused by this kind of problem. Times the cube on the origin of the problem, there are two myths and legends. The first attacks that plague Delos Island (Aegean island), a prophet said to have been the oracle of God, the altar of Apollo must be the volume doubling cube, can only stop the plague. A craftsman simply doubling each side of the altar (8 times the original volume into), which does not comply with God's will, so the plague is more rampant. Errors discovered, the Greeks will be the "Delian problem" to ask Plato. Plato said: God's real intention is to make the Greeks feel ashamed for ignoring geometry. Another story says that King Minos of Crete building tombs for his son, the command will be doubling the size of the original design, but still maintain a cubic shape. Angle trisection can not be because the real roots of cubic equation can not be made ruler. 5th century BC, Athens "Homo sapiens School" to the above questions centered research. Because of the ruler can not be used to solve, and often people go into new areas. For example inspired conic, circular curve, and cut three or four times the number found in algebraic curve. Since the establishment of the 17th century analytic geometry, the possibility of ruler and compasses only guidelines. 1837 PL Wang Wenzel given any angle and fold trisection can not use ruler and compasses cubic proof, 1882 CLFvon Lindemann proved the transcendence of π, square the circle can be established without possibility of . 1895 (C.) F. Klein summarizes the previous research, the "three geometric problems" (Chinese translation, 1930) a book, three major issues can not be given to mapping using simple ruler card method, solve the outstanding issues over two thousand years. . ∠ COB = 1 / 3 ∠ ACB The tools used here have been limited to the ruler, and the mapping methods with the Public sub. The other two problems can be resolved by other tools. The origin of geometry History of geometry (ie: "Geometry" is the name come from?) Produced in the same geometry and arithmetic practice, it can be said of history and arithmetic geometry produced is similar. In ancient times, people have accumulated in practice, very rich variety of flat, straight, square, round, long, short, paragraphs, narrow, thick and thin concepts, and gradually realize that between these concepts, they and their s position relationship between the number of relationships with the relationship between these later became the basic concepts of geometry. Production practices is the need, the original geometric concepts have gradually formed a relatively rudimentary knowledge of geometry. Although this knowledge is fragmentary, and most are empirical, but geometry is built on the fragmented, empirical, superficial knowledge of geometry above. Geometry is the branch of mathematics one of the oldest, but also in the field of mathematics that one of the most basic branch. Ancient China, Babylon, ancient Egypt, ancient India, ancient Greece is an important source geometry. A large number of archaeological finds prove that the prehistoric period in China, many people have mastered the basic knowledge of geometry, ancient times, people look at the items used in that many delicate, symmetrical pattern of the drawing, a number of simple design, but pay attention to vessel volume and volume ratio, are sufficient to show when people grasp how the geometric knowledge is enriched. Geometry is able to become a system of discipline, the Greek scholar who played a key role. Two thousand years ago, the commercial prosperity of ancient Greece, the production is more developed, a group of enthusiastic scholars, the pursuit of scientific knowledge, of geometry is the most interested in, should be mentioned here is the philosopher, geometer and philosopher Plato Aristotle's contribution to the development of geometry. Plato Logic introduced a geometric way of thinking, so that knowledge of the original geometry guided by the logic of the trend in the system and tight direction. In Athens, Plato taught his students to geometry, logical reasoning methods have been used in some of the propositions of geometry were demonstrated. Aristotle is recognized as the founder of logic, he's "syllogism" method of deductive reasoning, for the geometry of the development of even more enormous. Today, in elementary geometry, is still the use of syllogistic reasoning form. However, despite the time already had a very rich geometric knowledge, which is still fragmented, isolated, not systematic. Really summed up into a geometry with a relatively strict discipline theory is distinguished Greek mathematician Euclid. Euclid around 300 BC, Alexandria had to teaching, is a respected, Wenliang Dun thick educator. He loves math, know that some of the geometric principle of Plato. He is very thorough collection of the best known was the fact that all the geometry, according to Plato and Aristotle on logical reasoning methods, organized into a systematic theory has a strict, written in the early history of great works of mathematics - "Geometry." "Elements" of great historical significance is that it is axiomatic method to establish the earliest interpretation of the mathematical system model. In this work, the full knowledge of geometry are several assumptions from the first division, the use of logical reasoning methods to expand and narrative. In other words, from "Geometry" published in the beginning, geometry has truly become a relatively strict theoretical discipline system and the scientific method. Euclid's "Elements" Euclid's "Elements" Thirteen volumes, including Volume I talk about the conditions for congruent triangles, the triangle edges and corners of the size of the relationship between the theory of parallel lines, triangles and polygons equal area (area equivalent) conditions; the second volume about how the triangle into a square of equal area; third volume of round about; IV discuss the internal and circumscribed polygons; VI theory about similar polygons; fifth, seventh, eighth, ninth , Section X, and arithmetic was in about the proportion of; last about three-dimensional geometry of the content. From these elements can be seen that currently belongs to the secondary school curriculum in the elementary geometry of the main content has been fully included in the "Geometry" in the. So long time, people think that "Elements" is more than two thousand years, the standard transmission geometry textbook knowledge. Is "Elements" the content of geometry, Euclidean geometry, the people call it, or referred to as Euclidean geometry. "Elements" the most important feature is the establishment of more stringent geometric system, in this system has four main elements, definitions, axioms, postulates, propositions (including mapping and Theorem). "Elements" Volume I listed 23 definitions, five axioms, 5 postulates. (The last one is the famous parallel postulate postulate, or called the fifth postulate. It is the cause of the geometries in the history of over two thousand years the most famous on the "parallel line theory" discussion, and eventually gave birth to non-Euclidean geometry.) These definitions, axioms, postulates that "Elements" book basis. Book to these definitions, axioms, logically based on the public is _set_ to start his various parts. Each subsequent occurrence of such a theorem is stated in what is known and what is confirmation. Should be based on the previous definition, axioms, theorems for logical reasoning given careful proof. On the geometric method of argument, Euclid presented analysis, synthesis method and the reductio ad absurdum. The so-called analysis is the first assumption has been requested to analyze the conditions established at this time, so as to achieve proof steps; synthesis method is proven from the fact that before the beginning, and gradually exported to prove the matter; reductio ad absurdum is retain the proposition under the assumption of a negative conclusion, starting from the opposite conclusion, which export and has proven the facts and the known conditions of conflicting or contradictory results, thus confirming the conclusions of the original proposition is correct, also known as reductio ad absurdum. Euclid "Elements" of the birth history of the development in the geometry of great significance. It marked the geometry has become a theoretical system has a relatively strict and scientific methods of discipline. Released from the Euclid "Elements" to the present, two thousand years have passed, though of science and technology, but high school students to learn Euclidean geometry is still a good basic knowledge of mathematics teaching. Since Euclidean geometry has a distinct visual and logical deductive method has a strict combination of features in the long-term practice shows that it Pat-a culture, improve the green, young good teaching logical thinking ability. I do not know the history of the number of scientists to benefit from learning geometry, which made a great contribution. Newton's boyhood in Cambridge the night near the store to buy a book "Elements", started his view the content of the book is not out of common sense, and therefore did not read it carefully, while the Cartesian "coordinates geometry "is very interested in and concentrate on studying. Later, Newton in April 1664 to participate in special column in the scholarship exam, when Taiwan was unsuccessful, then the examiner, Dr. Barrow said to him: "Because you are too poor basic knowledge of geometry, no matter how hard is not enough." This I talk a big shock to the Newton. Thus, Newton again the "Geometry" from start to finish to repeated in-depth study, for future scientific work has laid a solid mathematical foundation. Development in the history of geometry, Euclid's "Elements" played a significant role in history. This role comes down to is presented geometrical "under" and the question of its logical structure. In his book, "Geometry" in the chain is the logical unfolding of getting from here to all the geometry, this work has not been done previous. However, in the long river of human knowledge, no matter how clever and famous predecessors, will not be completely solved the problem. Due to historical conditions, Euclid in the "Geometry" in the proposed geometry, "according to" problem has not been completely resolved, and his theoretical system is not perfect. For example, the definition of a straight line is an unknown fact to explain the definition of another definition of the unknown, such a definition can not play any role in logical reasoning. In another example, Euclidean logic used in the "continuous" concept, but in the "Geometry" has never mentioned the concept. Axioms of modern geometry People on the "Geometry" in there in the logical result of some of the holes, the discovery of flaws, it is to promote the constant development of the geometry of the opportunity. Finally, the German mathematician Hilbert summing up, based on previous work, published in 1899 in his "geometric basis," a book put forward a fairly complete system of axioms of geometry. The axiom system of Hilbert's axioms are called the body. Hilbert not only puts forward - a perfect geometric system, and also proposed the establishment of the principles of an axiomatic system. Is a geometric axiom systems, to which justice, should include how many axioms, should consider the following three areas: First, the coexistence (harmony), is an axiom system, the axiom should be no contradiction, and they coexist in harmony in the same system. Second, the independence of each axiom in the axiom system should be independent of each other and are not attached, there is no axiom can be derived from other axioms come. Third, completeness, contained in the axiom system of axioms should be enough to prove that the discipline of any new proposition. This axiom system used to define the basic geometry of the relationship between the object and its research methods, mathematics has become so-called "axiomatic method", and the Euclid in the "Elements" is called the classical axiom system proposed method. Axiomatic approach to geometry study brings a novel perspective, in justice law theory, because the basic object is not defined, and therefore do not have to explore the visual image of what the object, only the object of special study between the abstract relations, properties. Axiomatic method from the point of view, we can arbitrarily with points, lines, represent specific things, between things as long as they satisfy the axioms of the specific combination of relations, order relations, contractual relationship, so that these relations satisfy the axioms of the system requirements, which constitute the geometry. Therefore, any consistent axiomatic system could constitute the elements of geometry, the geometry of the visual image of each is more than just - a, but there may be infinitely many, each a visual image of the geometry we call it the explanation, or call a geometry model. Usually we are familiar with the geometry in the study of geometry, when not required, it is merely a visual image of it. In this regard, the geometry of the object of more extensive, and the meaning of geometry is more abstract than Euclidean times. These are the development of modern geometry has brought far-reaching impact.
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A world-famous in Elementary Geometry
As we all know, a triangle, if it is isosceles, then its two bottom corners of the angle bisectors are equal. A mathematical proposition true, people tend to like questioning the authenticity of its converse, and now Q: A triangle with two equal angle bisector, isosceles triangle if it is it? The answer is yes, but to prove that it is not so simple, the best method is to use reductio ad absurdum. Such as: In △ ABC the angle bisector BD, CD intersect at point D, BD = CD. Show that △ ABC is isosceles triangle. Certificate: ∵ BD = CD ∴ ∠ DBC = ∠ DCB ∵ BD, CD ∠ ABC and ∠ ACB is the angle bisector ∴ ∠ ABC = ∠ ACB ∴ AB = AC, that is △ ABC is isosceles triangle.
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Some well-known theorem of plane geometry
1, the Pythagorean Theorem (Pythagorean Theorem) 2, the projection theorem (Euclidean theorem) 3, the three middle triangle intersect at one point, and, by this point the center line into two parts of 2:1 4, the quadrilateral connecting both sides of the center and the center of the two diagonals intersect at one point connections 5, the interval of the edge connecting the center of the hexagon to the center of gravity of the two triangles coincide. 6, the vertical sides of a triangle bisectors intersect at one point. 7, the triangle of the three lines intersect at one point high 8, _set_ for the triangle ABC, circumcenter O, orthocenter for the H, O to BC from the lead edge of the vertical, _set_ the pedal for the L, then AH = 2OL 9, the triangle circumcenter, orthocenter, center of gravity in the same line (Euler line). 10, (nine round or circle or Euler Fair Bach circle) triangle, the triangular center, from the vertex perpendicular to its lead on the edges of the pedal, and orthocenter with the midpoint of the vertex, which nine points in a circle, 11, Euler's theorem: a triangle circumcenter, center of gravity, nine round heart, orthocenter turn in the same line (Euler line) 12, a large database on the theorem of Li Qi *: (cyclic quadrilateral of the nine round) Circle has four points, three points for the triangle than any of these four triangles of the nine round circle in the same heart, we had four heart nine round circle is called cyclic quadrilateral Round Nine . 13, (inside) of the three interior angle bisectors of a triangle intersect at one point, the radius of inscribed circle formula: r = (sa) (sb) (sc) s, s is half the perimeter of the triangle 14, (next to the heart) interior angle bisectors of a triangle and two other vertices of the exterior angle bisectors intersect at one point 15, central theorem: (Babs theorem) Let triangle ABC, side BC is the midpoint of P, then there AB2 + AC2 = 2 (AP2 + BP2) 16, Stewart Theorem: P will be within the triangle ABC, side BC is divided into m: n, there are n × AB2 + m × AC2 = (m + n) AP2 + mnm + nBC2 17, Bo Luomo and many theorems: cyclic quadrilateral ABCD the diagonal perpendicular to each other, the connection midpoint M and the diagonal intersection of AB line perpendicular to the CD E 18, Apollo Nice theorem: the two designated A, B be the ratio of the distance ratio of m: n (value not to 1) the point P, located in the line segment AB is divided into m: n of the points C and the outer points D is diameter of the circumference of both ends of the fixed point 19, Ptolemy Theorem: Let quadrilateral ABCD is inscribed in circle, there are AB × CD + AD × BC = AC × BD 20, an arbitrary triangle ABC, side BC, CA, AB as the bottom, respectively, are 30 degrees out for the bottom corner of the isosceles △ BDC, △ CEA, △ AFB, then △ DEF is an equilateral triangle, 21, Ireland, Marcos Theorem 1: If △ ABC and △ DEF is an equilateral triangle, by the line AD, BE, CF is the center of a triangle formed an equilateral triangle. 22, Ireland, Marcos Theorem 2: If △ ABC, △ DEF, △ GHI is an equilateral triangle, by the triangle △ ADG, △ BEH, △ CFI is the center of gravity of a triangle formed an equilateral triangle. 23, Menelaus Theorem: Let △ ABC triangular BC, CA, AB or its extension cord and one without which any intersection of the line vertices are P, Q, R, there BPPC × CQQA × ARRB = 1 24, the converse of Menelaus Theorem: (abbreviated) 25, Menelaus Theorem Theorem 1: Let △ ABC the exterior angle bisector of ∠ A cross-border CA in Q, ∠ C of the cross-border split line AB in R,, ∠ B the bisector of CA in the Q cross-border , then P, Q, R are collinear. 26, Menelaus Theorem Theorem 2: Over the three vertices of △ ABC arbitrary A, B, C as its circumcircle of the tangential, respectively, and BC, CA, AB of the extension cable at point P, Q, R, then P, Q, R are collinear 27, Ceva Theorem: Let △ ABC the three vertices A, B, C's not a triangle edge or point of their extended line connected surface S into three straight lines, respectively, with the side BC, CA, AB or their extension cords intersect at points P, Q, R, the BPPC × CQQA × ARRB () = 1. 28, Ceva Theorem Theorem: Let △ ABC is parallel to the side and both sides of the line BC AB, AC are the intersection of D, E, and _set_ up BE and CD intersect at S, then the AS would have had a side of the center of M BC 29, Ceva Theorem Inverse: (abbreviated) 30, the converse of Ceva's Theorem applied Theorem 1: The triangle intersect at one point the three middle 31, the converse of Ceva's Theorem application of Theorem 2: Let △ ABC the inscribed circle and the edges BC, CA, AB are tangent at point R, S, T, then the AR, BS, CT intersect at one point. 32, Ximo Song theorem: the circumcircle of △ ABC from any point P on the triangular BC, CA, AB or its extension cord for the vertical, _set_ the pedal are D, E, R, then D, E, R total line (this line is called Ximo Song Line) 33, Ximo Song inverse theorem: (abbreviated) 34, Steiner theorem: Let △ ABC's orthocenter for the H, the circumcircle of any point P, then the point P on the △ ABC's Ximo Song line through the center line PH. 35, Steiner Theorem Theorem: △ ABC's circumcircle point P on the side BC, CA, AB of symmetry points and △ ABC the same in a orthocenter H (with Ximo Song line parallel to) the line . This line is called the point P on the mirror line of △ ABC. 36, Bo Langjie, Teng following theorem: Let △ ABC's circumcircle of three points on the P, Q, R, then P, Q, R △ ABC intersect at one point on the necessary and sufficient conditions are: arc AP + BQ + arc arc CR = 0 (mod2Π). 37, Bo Langjie, Teng next Theorem Corollary 1: Let P, Q, R of the circumcircle of △ ABC is the three points, if P, Q, R △ ABC's Ximo Song on line intersect at one point, then A, B, C three of the Ximo Song on △ PQR lines intersect at the same point the previous 38, Bo Langjie, Teng next Theorem Corollary 2: In Corollary 1, three Ximo Song is the intersection of lines A, B, C, P, Q, R by the six appointed to take three points and the remaining three triangles orthocenter point of the triangle made by the midpoint of segment orthocenter connection. 39, Bo Langjie, Teng next Theorem Corollary 3: △ ABC's circumcircle test point on the P △ ABC's Ximo Song on the line, such as the _set_ QR is perpendicular to this line of the external pen Ximo Song strings the three points P, Q, R △ ABC's Ximo Song on line intersect at one point 40, Bo Langjie, Teng next Theorem Corollary 4: △ ABC from the vertex to the side BC, CA, AB cited vertical, _set_ the pedal are D, E, F, and _set_ side BC, CA, AB in the mid-point respectively, L, M, N, then D, E, F, L, M, N six in the same circle, then L, M, N points on the Ximo Song line on △ ABC intersect at one point. 41, a theorem about Ximo Song Line 1: △ ABC's circumcircle of the two endpoints P, Q lines on the triangle Ximo Song perpendicular to each other, the intersections in nine circle. 42, Theorem 2 on the line Ximo Song (peace Theorem): In a circle with 4 points to any of three points for a triangle, and then make the other point on the triangle Ximo Song line, these Ximo Song Line intersect at one point. 43, Carnot theorem: the circumcircle of △ ABC through the point P, and △ ABC cited the trilateral BC, CA, AB respectively, to the isometric into line with the PD, PE, PF, respectively, with the triangular intersection of D, E, F, then D, E, F are collinear. 44, Austria Bebel Theorem: The three vertices of △ ABC cited the three straight lines parallel to each other, _set_ them to the intersection of △ ABC's circumcircle, respectively L, M, N, taken in △ ABC's circumcircle point P, then PL, PM, PN and △ ABC triangular BC, CA, AB or its extension cord are the intersection of D, E, F, then D, E, F are collinear 45, Qing Theorem: Let P, Q is the circumcircle of △ ABC is different from A, B, C two points, P points on three sides BC, CA, AB respectively, the symmetric point of U, V, W, At this time, QU, QV, QW, and side BC, CA, AB or its extension cord are the intersection of D, E, F, then D, E, F are collinear 46, he took theorem: Let P, Q is the circumcircle of △ ABC on a pair of counter-point, point P on the three sides BC, CA, AB respectively, the symmetric point of U, V, W, then, if QU , QV, QW and the side BC, CA, AB or its extension cord are the intersection of ED, E, F, then D, E, F are collinear. (Anti point: P, Q, respectively, the radius of the circle O OC and its extended line of two points, if the OC2 = OQ × OP called P, Q two points against each other on the circle O points) 47, Long since ancient times theorem: in the same circle above a A1B1C1D14 points to any of three points for the triangle in the circle take a little P, for P points on the 4 triangles Ximo Song line, then from P to the four Ximo Song vertical line of argument, the four foot drop on the same line. 48, nine circle theorem: the midpoint of sides of a triangle, three high point of the Foot and three Euler [link vertices of a triangle from three line segments with the midpoint of orthocenter] A total of nine round [often called the circle of the Kowloon point circle [nine-point circle], or Euler circle, Feuerbach circle. 49, a circle with n points, from which any n-1 points of focus in the rest of the circle to the tangent point on the vertical are cited intersect at one point. 50, Cantor Theorem 1: A circle with n points, from which an arbitrary n-2 points of focus of the remaining two points of connection to the vertical cited a total of points. 51, Cantor Theorem 2: A circle with A, B, C, D four-point and M, N points, then M and N points on the four triangles △ BCD, △ CDA, △ DAB, △ ABC in Each intersection of the two Ximo Song along the same line. This line is called M, N points on the quadrilateral ABCD of Cantor line. 52, Cantor Theorem 3: A circle with A, B, C, D four-point and M, N, L three points, then M, N points on the quadrilateral ABCD of Cantor line, L, N two point line on the quadrilateral ABCD of Cantor, M, L points of the Cantor on line quadrilateral ABCD intersect at one point. This point is called M, N, L Cantor three points on the quadrilateral ABCD's. 53, Cantor Theorem 4: A circle with A, B, C, D, E five points and M, N, L three points, then M, N, L three points on the quadrilateral BCDE, CDEA, DEAB, EABC in Cantor points each in a straight line. This line is called M, N, L three on the Pentagon A, B, C, D, E of the Cantor line. 54, Fair Bach Theorem: The Nine Circles and triangles inscribed circle and the inscribed circle tangent to the side. 55, Morley Theorem: The triangle's three angles of three equal, close to the edge of a phase to get a two-thirds moldings intersection, the intersection of these three can form a triangle. This triangle is often called the Morley triangle. 56, Newton's Theorem 1: Quadrilateral two pairs of side extension cable connecting the intersection of the diagonal section of the midpoint and the midpoint of the two, three collinear. This line is called the Newton line of the quadrilateral. 57, Newton's Theorem 2: The circle circumscribed quadrilateral the two diagonals of the midpoint, and the center of the circle, three points are collinear. 58, flute Shag Theorem 1: The plane has two triangles △ ABC, △ DEF, _set_ their corresponding vertices (A and D, B and E, C and F) connections intersect at one point, then if the corresponding edge or extend the lines cross, then the three intersection points are collinear. 59, flute Shag Theorem 2: There are two different triangle plane △ ABC, △ DEF, _set_ their corresponding vertices (A and D, B and E, C and F) connections intersect at one point, then if intersection of corresponding sides or the extension cord, then the three intersection points are collinear. 60, Wembley Anson Theorem: link external hexagon ABCDEF inscribed in the circle opposite vertex A and D, B and E, C and F, the total of the three point line. 60, Pascal theorem: hexagon ABCDEF inscribed opposite sides AB and DE, BC and EF, CD and FA (or extension cord) intersection are collinear.
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Geometric view of the world's celebrities
1. Believe that algebra is real. ---- Gaussian (Gauss) 2. Shuxingjiege, a small number of short form direct and form gaps of difficulty of nature. ----- Hua 3. "Experiment failed to treat our scientists Wang Juzhen proverb, called" stay here with 50% chance of success, none of that is 100% failure. " ------ Wang Juzhen 4. "A man is like a score, he can be like the actual molecule, and the valuation of his own like denominator. Denominator is larger, the smaller the fraction of the value." Tolstoy ----- 5. "The essence of mathematics lies in its freedom ."---- Hong Alto, Seoul (Cantor) 6. "In the field of mathematics, the art of asking questions than answering the question of art is more important ."---- Hong Alto, Seoul (Cantor) 7. "No problem can be so deeply moved by the endless human emotion, few other concepts such as infinite as incentives to produce fruitful rational thinking, but there is no concept of any other need to be clarified as to the infinite ."---- Hilbert (Hilbert) 8. "Mathematics is the science "---- infinite Hermann Weyl 9. "The problem is the heart "---- PRHalmos 10. "Branch of science can be as long as a large number of issues raised, it is full of vitality, and the question indicates the lack of independent development of the termination or decline." ---- Hilbert 11. "Some of the beautiful theorems in mathematics have such features: they are easily summarized from the facts out, but the proof is hidden deep ."---- Gaussian 12. "Time is a constant, but the hard-working person, it is a 'variable'. With a 'points' to calculate the time than with the 'h' to calculate the time of the person 59 times more time." ---- Reba Clain 13. "Courage in learning to subtract, that is, minus some of their predecessors have been resolved, there are those who see the problem is not solved, we need to explore solutions." Hua ---- 14. "Days = 2% inspiration and +98% of the blood and sweat 。"---- Edison 15. "To make use of time to think about doing the day is 'positive sign' or 'negative', if a '+', then progress; If it is '-', you have to learn to take measures . "---- Dimitrov 16. "Greatest modern scientist Albert Einstein when talking about success, write a formula: A = x + y + z. And explained: A representative of success, x behalf of hard work, y representative of the right way , Z on behalf of less empty talk. "---- Albert Einstein 17. "Some of the beautiful theorems in mathematics have such features: they are easily summarized from the facts out, but the proof is hidden deep. Mathematics is the science of the king." ---- Gauss 18. "In the field of mathematics, the art of asking questions than answering the question of art is more important." ---- Kang Alto, Seoul 19. "Branch of science can be as long as a large number of issues raised, it is full of vitality, but the problem is the lack of independent development indicates the termination or decline." ---- Hilbert 20. "In the world of mathematics, the important thing is not what we know, but how do we know something." ---- Pythagoras 21. "A science only when it successfully applied math in order to achieve a real good point." ---- Marx 22. "Scientific standards of a country can use it to measure the consumption of mathematics." Rao ---- 23. "Mathematics - the unshakable foundation of science, and promote a rich source of human progress in the cause." ---- Barrow 24. "In the Mount Olympus ruled by God, but the number of the eternal." ---- Jacobian 25. "If there are no numbers on the manufacture of imitations of eternal universe, the humans will not survive." ---- Nietzsche 26. "Do not know who to avoid feed geometry." ---- Plato 27. "Geometry without the King of the Road!" ---- Euclidean 28. "Mathematician who is actually a fascinating, no fans, no mathematics." Novalis ---- 29. "Bold speculation is not to not make great discoveries." ---- Newton 30. "Ruled the universe 。"---- Pythagorean numbers 31. "Mathematics, science, Queen; number theory, mathematical Gaussian Queen 。"---- 32. "God made the integers, all the rest of the numbers are artificial." ---- Kelongneike 33. "God is an arithmetic Home" ---- Jacobian 34. "Popular poetry without a sort of mathematician never become a complete mathematician 。"---- Weierstrass 35. "Branch of science, pure mathematics, then the modern stage of development of the human spirit can be said that the most original creation 。"---- Whitehead 36. "To the whole number is an amount of ruling the world, and counts four operations can be seen as all the equipment 。"---- mathematician Maxwell
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English Expression
: geometry, adj of geometry, of or like the lines, figures, etc used in geometry, geometric
n.: quadrivial, quadrivium, how much; how many, how much
euclidean