A square on the side. As the picture shows, we
Using hook (a) and chord (b) to represent right-angled triangles respectively to get two right-angled sides, and using chord (c) to represent hypotenuse, we can get: hook 2+ chord 2= chord 2.
Namely: a2+b2=c2. This is the most important theorem in geometry and is widely used. According to China's ancient mathematical masterpiece Nine Chapters Arithmetic, Pythagorean Theorem was discovered by Shang Gao of Zhou Dynasty thousands of years ago, and was later annotated by Zhao Shuang of Han Dynasty.
Therefore, in our country, Pythagorean theorem is also called "quotient height theorem". In western countries, Pythagorean theorem is called "Pythagorean theorem", but Pythagoras discovered this theorem much earlier than the quotient in China.
It's late
At the beginning of China's earliest mathematical work "Parallel Calculation of Classics in Weeks", there was a dialogue in which the Duke of Zhou asked Shang Gao for mathematical knowledge:
Duke Zhou asked, "I heard that you are very proficient in mathematics. Excuse me: there is no ladder to go up in the sky and no ruler to measure on the ground. So how can we get data about heaven and earth? "
Shang Gao replied: "The number comes from the understanding of the other party and the circle." There is a principle: when the moment of a right triangle gets a right-angled side' hook' equal to 3 and the other right-angled side' chord' equal to 4, then its hypotenuse' chord' must be 5. This truth was summed up when Dayu was in charge of water conservancy. "
From the above conversation, we can clearly see that people in ancient China discovered and applied the pythagorean theorem, an important principle of mathematics, thousands of years ago. [Return]
Interesting Pythagorean Theorem
Greece issued a stamp on 1955. The design consists of three chessboards. This stamp commemorates the Pythagorean School, a school and religious group in Greece 2,500 years ago, its establishment and its cultural contribution. The design on the stamp is an explanation of a very important theorem in mathematics. It is the most wonderful, famous and useful theorem in elementary geometry. In our country, people call it Pythagorean theorem or quotient height theorem; In Europe, people call it Pythagoras theorem.
Pythagorean theorem asserts that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. If we want to find a theorem, its appearance can be called a milestone in the history of mathematical development, then Pythagorean theorem is the best choice. However, if people want to study the origin of this theorem, they will often be confused. Because in Europe, people attribute the proof of this theorem to Pythagoras; However, by studying the cuneiform clay tablets unearthed in Mesopotamia in the 20th century, it is found that the Babylonians knew this theorem more than a thousand years before Pythagoras. In the astronomical almanac of the Western Han Dynasty or earlier in China, the first chapter describes the questions and answers between Shang Yang and Zhou Gong Ji Dan during the establishment of the Western Zhou Dynasty (about 1000 BC). Duke Zhou asked Shang Gao, "The sky can't rise in steps, and the land can't be measured in inches." How are some measurements of sky and ground height obtained? Shang Gao replied, "So the moment of folding is three, four and five." That is, we often say tick three, divide four and string five. It is also recorded in the Weekly Calendar that the weekly calendar is eight feet long and the summer sundial is one foot six inches long. Beard, stock, upright, hook. A thousand miles due south, one foot five inches, a thousand miles due north, one foot seven inches. The south is getting longer and longer. Wait six feet, that is, take bamboo, one inch and eight feet long, catch a picture and watch it, the room will cover the sun, and the day should be empty. From this point of view, the velocity is 80 inches and the diameter is 80 inches. Therefore, the hook is the head and the moustache is the strand. From the moustache to the sun, it is 60 thousand Li, without a shadow. It's 80 thousand Li from now on.
This passage describes how ancient people in China used Pythagorean theorem to practice in science. Professor Qian Weichang explained this passage in detail: "... Shang Gao and others measured the shadow of the sun with a vertical rod (namely Zhou Xie), and then calculated the height of the sun with Pythagoras method. Zhou Xie is eight feet tall. In Haojiang (near Xi 'an today), the sun's shadow from summer solstice is one foot six inches long and one foot five inches due south. Thousands of miles due north, the shadow is one foot and seven inches long. By measuring the shadow of the sun, the ancestors skillfully calculated the inclination of the sun from the ground in summer to the sun, and similarly measured the inclination of the sun from winter to the sun. We also took a hollow bamboo tube with a diameter of one inch and a length of eight feet, and used it to observe the sun. Our ancestors found that the circular shadow of the sun just filled the line of sight of the bamboo tube, so they calculated the diameter of the sun according to the oblique height of the sun and the Pythagorean principle. Although these measured data are very rough and far from the actual situation, we should learn from the creative and practical observation spirit of such a genius as early as 3000 years ago. "Therefore, it is completely reasonable for China people to call this theorem Pythagorean Theorem or Quotient Theorem.
But Europeans call this theorem Pythagoras theorem, and they also have their own opinions. Because Pythagoras himself, at least a member of Pythagoras school, first gave the logical proof of this theorem. Although Pythagoras has many outstanding proofs, such as proving that √2 is not a rational number by reduction to absurdity, the most famous proof is Pythagoras theorem. Legend has it that when he got this theorem, he was very happy and killed a cow as a sacrifice to the gods. Some historians say it's a hundred cows, which is too expensive!
Pythagorean theorem is the theorem with the most proving methods in mathematics-there are more than 400 explanations! The proof method displayed on Greek stamps was first recorded in Euclid's Elements of Geometry.
Zhao, a mathematician in the Han Dynasty, attached a picture to prove the Pythagorean theorem when he annotated the Classic of Parallel Computing in the Zhou Dynasty. This proof is the simplest and most ingenious explanation of more than 400 Pythagorean theorems. Can you find out how Miss Zhao proved this theorem? (Hint: Consider the area calculation of the square with black edges) [Return]
Superior Pythagorean Theorem
Based on Pythagorean Theorem introduced in textbooks, we can further understand the discovery, proof and application of Pythagorean Theorem through the Internet. From the vivid historical data of mathematics, we know that China had a splendid culture in ancient times and formed a splendid mathematical culture in the field of mathematics. At least twenty or thirty mathematical achievements, such as Pythagorean Theorem, were once in the leading position in the world.
First of all, the earliest one of China's top ten famous computing classics, Zhou Kuai Shu Jing. The book records "Gou Guangsan, Gu, Wu" as a special case of Pythagorean theorem, which prepares for the formation of Pythagorean theorem. The Pythagorean Theorem is described more brilliantly in The Classic of Weekly Parallel Computing: "If you seek evil from the sun, take the sun as a hook, the sun will gather, and the hook and the shares will be multiplied separately, and you will get evil to the day." The general Pythagorean theorem has been involved. Formulated, the chord (evil to the sun) is equal to the square sum of the square of the hook plus the square of the strand. It can be seen that China has independently discovered Pythagorean theorem.
Secondly, from the proof method of Pythagorean theorem, we have received effective patriotism education. The textbook 1 in this chapter introduces three methods of proof. "Let's broaden our horizons and feel how ingenious and simple it is for Zhao Shuang, an ancient mathematician in China, to prove Pythagorean theorem (question P225, 12) with Pythagorean square diagram." According to the string diagram, we can also multiply Pythagoras into Zhu Shi and then double it into Zhu Shi. Take the Pythagorean difference as the middle yellow solid, and add the difference to become the chord solid. "Write it with the formula: 2ab+(b-a) 2 = C2+B2 = C2. Combining geometry knowledge with algebra knowledge can be described as "original creation". What a novel and wonderful mathematical method it was in ancient China! Nowadays, there are still many mathematical problems in the world, waiting for us to overcome and fill them, and use our diligence and wisdom to pick off the pearls of mathematics.
Through the introduction of these vivid digital historical materials, our enthusiasm for learning suddenly rose, and we are all proud of our motherland's brilliant achievements! Patriotic enthusiasm arises spontaneously! This not only enables us to receive patriotic education, but also enables us to understand Pythagorean theorem more profoundly from vivid historical materials.
The organic combination of mathematics philosophy, mathematics history and mathematics teaching has become a hot issue in the world today.
In the process of studying Pythagorean theorem and searching information on the Internet, we also thought of Zu Chongzhi in ancient China, and got the approximate value of л, which was accurate to the seventh place after the decimal point, leading the world 1000 years. Liu Hui's graph cutting technique, the pioneering "Wide Seeking Technique" and "Yang Hui Triangle", as well as the great achievements of contemporary famous mathematicians such as China, Soviet Union and Chen Jingrun and their patriotic feelings of winning glory for our country. [Return]
Ancient mathematicians in China proved Pythagorean theorem.
Mathematicians in ancient China not only discovered and applied Pythagorean Theorem very early, but also tried to prove Pythagorean Theorem in theory very early. Zhao Shuang, a mathematician of the State of Wu in the Three Kingdoms period, was the first to prove the Pythagorean theorem. Zhao Shuang created "Pythagorean Square Diagram" and gave a detailed proof of Pythagorean theorem by combining shape and number. In this Pythagorean Square Diagram, the square ABDE with the chord as the side length is composed of four equal right triangles and a small square in the middle. The area of each right triangle is AB/2; If the side length of a small square is b-a, the area is (b-a)2. Then you can get the following formula:
4×(ab/2)+(b-a)2=c2, which can be obtained after simplification:
A2+b2=c2, that is, c=(a2+b2)( 1/2).
Zhao Shuang's proof is unique and innovative. He proved the identity relationship between algebraic expressions by cutting, cutting, spelling and supplementing geometric figures, which was rigorous and intuitive, and set a model for China's unique ancient style of proving numbers by shape, unifying numbers by shape, and closely combining algebra and geometry. Later mathematicians mostly inherited this style and developed it from generation to generation. For example, Liu Hui later proved Pythagorean theorem by means of formal proof, but the division, combination, displacement and complement of specific numbers are slightly different. The discovery and proof of Pythagorean theorem by ancient mathematicians in China has a unique contribution and position in the history of mathematics in the world. In particular, the thinking method of "unity of form and number" embodied in it is of great significance to scientific innovation.
Figure 2 Pythagorean Square Diagram In fact, the thinking method of "unity of form and number" is an extremely important condition for the development of mathematics. As Wu Wenjun, a contemporary mathematician in China, said, "In China's traditional mathematics, the relationship between quantity and spatial form often develops side by side ... Descartes invented analytic geometry in the17th century, which is the reappearance and continuation of China's traditional thoughts and methods after hundreds of years of pause." [Return]
The President of the United States skillfully proved the Pythagorean theorem.
Anyone who has studied geometry knows Pythagorean theorem. It is an important theorem in geometry and is widely used. So far, there are more than 500 ways to prove Pythagorean theorem. Among them, Garfield, the twentieth president of the United States, had a story in the history of mathematics.
Why did the president think of proving Pythagorean theorem? Is he a mathematician or a math lover? The answer is no. Here's the story.
1876 One weekend night, on the outskirts of Washington, DC, a middle-aged man was walking and enjoying the beautiful scenery in the evening. He was a party member in * * * Ohio and Garfield. Walking, he suddenly found two children talking about something on a small stone bench nearby, arguing loudly and discussing quietly. Curious, Garfield followed the sound and walked to the two children. I wonder what these two children are doing. I saw a little boy bend down and draw a right triangle on the ground with branches. So Garfield asked them what they were doing. The little boy said without looking up, "Excuse me, sir, if the two right angles of a right triangle are 3 and 4 respectively, what is the length of the hypotenuse?" Garfield replied, "It's 5." The little boy asked again, "If the two right angles are 5 and 7 respectively, what is the length of the hypotenuse of this right triangle?" Garfield replied without thinking, "The square of the hypotenuse must be equal to the square of 5 plus the square of 7." The little boy added, "Sir, can you tell the truth?" Garfield is speechless, unable to explain, and has a bad psychology.
So Garfield stopped walking and immediately went home to discuss the problems left by the little boy. After repeated thinking and calculation, he finally figured it out and gave a concise proof method.
He analyzed it like this, as shown in the figure:
On April 1876, Garfield published his proof of Pythagorean theorem in the New England Journal of Education.
188 1 year, Garfield became the twentieth president of the United States. Later, in order to commemorate his intuitive, simple, understandable and clear proof of Pythagorean theorem, people called this proof "President". Proof method.
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Generalization of Inverse Theorem
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From Pythagorean Theorem to Fermat's Last Theorem
Cao Daomin, Deputy Director of Institute of Applied Mathematics, China Academy of Sciences
Many people in their thirties have heard of Goldbach's conjecture, because mathematicians in China have made outstanding contributions to this conjecture, especially Chen Jingrun's result is the best. Chen Jingrun's deeds were widely circulated throughout the country in 1980s, which influenced many young people at that time. Now people in their forties who are engaged in mathematical research, including myself, have been influenced and embarked on the road of scientific research.
If someone asks what is the most important achievement of mathematics in the last century, I believe many people will say Fermat's last theorem. This puzzle, which has been hanging for more than 350 years and is more famous than Goldbach's conjecture, was completely solved by the British mathematician wiles in 1995. 1In March 1996, Wells was awarded the Wolf Prize.
First, let's introduce Fermat's last theorem.
Anyone who has studied plane geometry knows that if A and B are two sides of a right triangle, then the length C of the hypotenuse and A and B satisfy the relation c2 = a2+b2. China called it "Shang Gao Theorem" because it was recorded in the ancient mathematical work "Zhou Kuai Shu Jing" that the ancient mathematician Shang Gao talked about this relationship. More generally, it is also called Pythagorean theorem. This is because "Hook 3, Strand 4 and Chord 5" are recorded in Zhouyi Suan Jing, and their relationship with right triangle is clearly discussed. There are other Pythagorean numbers in subsequent works. For example, there are seven groups of numbers (5, 12, 13), (7, 24, 25), (8, 15, 17) in Nine Chapters of Arithmetic. In the west, the above formula is called Pythagorean Theorem, because western mathematics and science originated in ancient Greece, and the oldest work made in ancient Greece is Euclid's Elements of Geometry, and many of these theorems naturally fall on Pythagoras, knowing that Pythagoras is regarded as the "father of number theory".
If the unknowns of A, B and C in Pythagorean Theorem c2 = a2+b2 are the first indefinite equation (that is, the number of unknowns is more than the number of equations) and the first indefinite equation to get a complete solution, on the one hand, it leads to various indefinite equations, on the other hand, it also establishes a paradigm for solving indefinite equations.
Pierre de Fermat (160 1- 1665), a Frenchman, was interested in mathematics, although he studied law and was a lawyer. He often reads math books in his spare time and does some math research by himself. When reading the book Arithmetic by the Greek mathematician Dio Fentos, the general solution of x2+y2 = z2 was discussed. In the margin of the book, he wrote down his experience with a pen: "Conversely, it is impossible to split a cubic number into the sum of two cubic numbers, and it is impossible to split a square number into the sum of two square numbers. More generally, any square number greater than two cannot be divided into the sum of two identical squares. I found a wonderful proof, but the blank is too small to write the whole proof. " Expressed in mathematical language, Fermat concluded:
When n≥3, xn+yn = zn has no positive integer solution.
People don't believe that Fermat found the proof of this conclusion, or, like thousands of future generations, think that he has proved it but is actually wrong, because many famous mathematicians tried to prove it, but all ended in failure. However, Fermat did create the infinite descent method and proved that n = 4. The case of n = 3 is given by Leonard Euler (1707-1783) in 1753. In fact, at the beginning of19th century, only two cases of n = 3 and n = 4 were proved. The case of n = 5 was completely proved for the first time after more than half a century until 1823 to 1825. Fermat's last theorem was the greatest challenge faced by mathematicians at that time. In order to show that academic circles attach importance to it, the French Academy of Sciences set up the Fermat Last Theorem Award for the first time in 18 16. Many great mathematicians, including the top mathematicians at that time, Gauss of France and Cauchy of France, were keen on this problem.
Among the heroes who tried to solve Fermat's Last Theorem in the early days, there was a heroine, German sophie germain (1776- 183 1). When she was a child, she was a shy and timid girl who taught herself math and read books. Because the female surname was discriminated against in mathematics at that time, she used a male pseudonym to correspond with some great mathematicians, including Gauss and Legendre. Her talent surprised these first-class mathematicians.
Now let's look back at Pythagorean theorem.
a2 + b2 = c2
If we divide by c2 on both sides of the equation, we get
= 1
Let = x, = y, then find positive integers A, B, C satisfying a2+b2 = c2, and find rational numbers X, Y equivalently, so that (X, Y) satisfies x2 +y2 = 1. (x, y) can be regarded as a point on the unit diagram on the plane, and the point (x, y) where both x and y are rational numbers is called a rational point. In this way, whether there is a positive integer in the equation obtained by Pythagorean theorem can be solved as whether there is a rational point on the unit circle on the plane. Similarly, whether xn+yn = zn has a positive integer solution is equivalent to whether there is a rational point on the curve xn+yn = 1 on the plane. We call the curve defined by the equation xn+yn = 1 Fermat curve.
In middle school mathematics, we have some knowledge about plane algebraic curves. In analytic geometry, we have completely classified quadratic curves. The quadratic algebraic curve on the plane is
Ellipse:;
Hyperbola:, or;
Parabola:
Algebraic geometry plays a very important role in solving Fermat's last theorem. Algebraic geometry is a natural continuation of analytic geometry. In analytic geometry, we use coordinate method to represent curves and surfaces through equations. Usually we only study linear and quadratic curves, that is, straight lines, ellipses, hyperbolas and parabolas. Cubic and cubic curves are generally not carefully studied.
One of the main differences between algebraic geometry and analytic geometry is that analytic geometry uses degrees to classify curves and surfaces, while algebraic geometry uses a doubly rational transformation invariant-genus to classify algebraic curves. Through genus G, all algebraic curves can be divided into three categories:
G=0: straight line, ellipse and conic curve;
G= 1: elliptic curve;
Other curves, especially Fermat curves.
The genus of Fermat curve is a famous conjecture put forward by British mathematician Mo Deer in 1929, that is, there are only a limited number of points on the algebraic curve of genus. In 1929, Siegel proved that there are only finite integer points on the algebraic curve of genus.
Of course, there are many more rational numbers than hours.
1983, the German mathematician Fielding proved Mo Deer's conjecture. His proof uses the results of many mathematicians. His result is regarded as a great theorem in the last century, for which he won the Fields Prize of 1986. Starting from Mo Deer's conjecture, we deduce that if xn+yn = zn has coprime nontrivial positive integer solutions, then the number of solutions is limited. Heath-Brown used Mo Deer conjecture to prove that Fermat's Last Theorem holds for almost all prime numbers.
Because of the proof of Mo Deer's conjecture, mathematicians have seen that a series of conjectures can eventually lead to the proof of Fermat's last theorem.
1983, Lucien Szpiro put forward the Spiro conjecture, and proved that Fermat's last theorem can be deduced from the Spiro conjecture, and it holds for a sufficiently large exponent. 1985, he and D.W.Masser put forward a series of equivalent conjectures, one of which is called abc conjecture, from which Spiro conjecture can be deduced. 1987, spiro put forward a series of conjectures, from which spiro conjecture can also be deduced. These conjectures seem easier to start, but none of them have been proved so far.
1987, Searle put forward some stronger conjectures, which are called Searle strong (weak) conjectures. Not only Fermat's last theorem can be deduced from it, but also many other conjectures can be deduced, but this road finally failed.
197 1 year, Legua was the first person to connect the elliptic curve with Fermat's last theorem, but Gerhard Freddy was the first person to turn the direction to the right track. In 1985, Frey proved that if the Fermat equation (a prime number not less than 5) has a non-zero solution (that is, an elliptic curve can be designed, it can be assumed to be a coprime non-zero integer, and obviously it is an elliptic curve in the rational number field.
Japanese mathematician Yutaka Taniyama (1927 ——1958) studied the parameterization of elliptic curves at the meeting held in 1955. Parameterization of a curve is very helpful for the expression of curve and the study of curve properties, which we have seen when we studied analytic geometry in middle school. Elliptic curve is cubic curve, which can also be expressed by some functions. However, if the function used in parameter representation can be modular (modular function is a kind of meromorphic function everywhere on the upper half complex plane, and modular form is a generalization of modular function), then we call it modular curve. The modulus curve has good properties. We hope that any elliptic curve is a modulus curve, which is Murakami's guess. Since then, mathematicians have turned proving Fermat's Last Theorem into proving that the Gu Shan Yi Cun conjecture is true for a certain kind of elliptic curve.
It is along this road that the British mathematician Wells finally made a breakthrough in1June, 993 after seven years of long exploration. Finally, Fermat's Last Theorem was completely proved in 1995.
At the end of this paper, I would like to give some advice to math lovers: there are some seemingly simple conclusions in mathematics, such as Goldbach conjecture and Fermat's last theorem, which are very difficult to prove. Many math lovers think that with good "inspiration", they can solve world problems with elementary math methods or few math tools, and as a result, they waste a lot of precious time. Recently, I often read in newspapers and online that XXX has solved the problem of XXX, and some irresponsible reports in the media may mislead some math lovers. Let readers know the solution process of Fermat's Last Theorem, and hope that math lovers will not blindly solve world problems, which is also one of the original intentions of this paper. If you really love mathematics and are determined to solve mathematical problems, you should first learn the basic knowledge of a certain major, understand the international research trends of this problem, understand the work of predecessors, and then conduct your own research.
The writing of this article refers to Professor Hu Zuoxuan's "From Pythagoras to Fermat" and "A Journey of 350 Years-From Fermat to Wells", and I would like to express my gratitude here. Since my major is not number theory, I am likely to make mistakes in the article. Please correct me. Readers who want to know more can look at these two books by Professor Hu Zuoxuan. )
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Application of Pythagorean Theorem
Pythagorean theorem is one of the important theorems in junior middle school mathematics. It reveals the quantitative relationship among the three sides of a right triangle, which can solve many problems in the calculation and proof of right triangles. It is one of the main bases for solving the problems of right triangles and is of great use in production and life. Therefore, it is a problem that junior high school mathematics should pay attention to and must be solved, and we should have a deep understanding and extensive understanding of it.
First, pay attention to initial learning, understand the history of theorem discovery, and stimulate interest in learning.
In the initial study, according to the content of the textbook and the history of China's mathematics development, I learned about the achievements in the study of ancient Pythagorean Theorem in China, which stimulated my thoughts and feelings of loving the long-standing culture of the motherland and cultivated national pride. At the same time, combined with many scientific examples in the world today, it has aroused our interest in learning mathematics, inspired ourselves to work hard and study hard, and laid a solid foundation for shouldering the heavy responsibility of rejuvenating China in the future.
Example 1 Many scientists in the world today are trying to find "people" on other planets, and have sent many signals to the universe, such as human language, music, various graphics and so on. It is said that Hua, a famous mathematician in China, once suggested that the graph of Pythagorean theorem should be deduced. If cosmic people are "civilized people", then they will certainly understand this "language". Do you think this is possible?
Second, pay attention to the analysis of theorem structure and correctly understand and apply theorems.
Before learning Pythagorean Theorem, although we already have some basic knowledge such as right triangle and proposition, after understanding Pythagorean Theorem, we should not only analyze the theorem structure to make ourselves understand it correctly, but also apply the theorem or its simple deformation to solve problems accurately and extend it to extracurricular activities.
Example 2 Try to rewrite the proposition "The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse" as "If …, then …".
Example 3 Find the height of an equilateral triangle with side length A. 。
ask
The length of (1)AD.
(2) the area of △ Abd.
Example 2 aims to make us clear about the theme and conclusion of Pythagorean theorem so as to apply it correctly.
After drawing the diagram, it is obvious that Pythagorean theorem can be directly applied to solve it.
Example 4 needs to be calculated by using the simple deformation b2=c2-a2 of Pythagorean theorem.
Third, improve it in time and apply the theorem flexibly.
In the research process of Pythagorean theorem application, we can choose a slightly more difficult example to train our flexibility in applying the theorem.
Example 5 As shown in Figure 2, AD is the height on the BC side of △ABC.
Verification: AB2+CD2=AC2+BD2
It shows that when analyzing this problem, we should first consider that there are two right-angled triangles with common sides in the graph-△ABD and △ACD. The four line segments AB, CD, AC and BD are all in the form of squares, which are the hypotenuse and right-angled sides of △ Abd and △ACD respectively. Obviously, it is necessary to apply Pythagorean theorem to concentrate the related sides of the same right triangle.
Fourth, pay attention to the exploration and learning after the application of the theorem, and give new methods to the application of the theorem in time.
After applying the proof of clear understanding, it is not only beneficial to cultivate our ability and creativity in analyzing problems, but also makes the application of Pythagorean theorem reverberate endlessly.
Example 6 at △ABC, ∠ c = 90, verify sin2A+sin2B= 1.
It shows that when solving a right triangle, if two of its three sides are known, the length of the third side can be obtained by Pythagorean theorem. After solving the problem, it is not difficult to find that by combining Pythagorean theorem with acute trigonometric function, it can be proved that sin2A+sin2B= 1.
Example 7 is shown in fig. 3. In △ABC, AB=AC, and D is any point on the side of BC. Proof: AB 2-Ad2 = BD DC.
It shows that although this example does not meet the Pythagorean theorem as in Example 5, if we further study the proof method of Example 5 in combination with the conclusion of this example, it is not difficult to find that this example also conforms to Pythagorean theorem (if BD DC = m2, the formula to be proved is AB2-AD2 =m2).
When analyzing this problem, we should consider the square form of the line segment in the verification conclusion, and we can use Pythagorean theorem. But because there is no right triangle in the picture, we need to add a vertical line to construct a right triangle. Because there are AB2 and AD2 in the verification, we must form a right triangle with AB and AD, so we need to make AE⊥BC in E. As shown in Figure 3, we can deduce from Pythagorean theorem:
How to launch BD DC from BE2-ED2? This requires the use of the square difference formula.
BE2-ED2=(BE+ED)(BE-ED)，
And BE+ED=BD,
From the nature of isosceles triangle, we can know that BE=CE, and then we can deduce that BE-DE=CD.
So be2-ed2 = BD DC, so the problem is proved.
Fifth, combine the theorem and its inverse theorem to deepen the understanding and application of the theorem.
Pythagorean theorem and its inverse theorem reflect the relationship between property theorem and judgment theorem. Correctly distinguishing Pythagorean theorem and its inverse theorem can further deepen our understanding of the nature and judgment relationship of right triangle. In the process of study and research, when to use the theorem and when to use the inverse theorem, especially the Pythagorean theorem, can not only deepen our understanding of Pythagorean theorem, but also broaden our horizons, broaden our knowledge and understand various methods in mathematics.
Let n be a natural number in Example 8, and prove that the triangle with 2n2+2n, 2n+ 1, 2n2+2n+ 1 as sides is a right triangle.
Illustrative example 8 mainly triangulates Pythagorean theorem in the form of a2+b2=c2.
Example 9 should be analyzed according to Pythagorean theorem before it can be put into practice.
Sixth, explore the proof of theorem and broaden the application of theorem.
There are hundreds of ways to prove Pythagorean theorem in the world. Although Pythagorean theorem has been proved in textbooks, in the process of research, if we can properly understand and learn some proofs outside textbooks according to our own abilities, it will not only be conducive to the application and understanding of the theorem, but also enable us to find new ways to solve problems and cultivate our creative thinking ability.
Example 10 The following is a method to prove Pythagorean theorem by using a graph. Try to explain how to prove Pythagorean theorem according to the given figure. Can other divisions be designed to prove Pythagorean theorem?
Note: The study of example 10 not only enables us to master a variety of methods to prove problems and cultivate our thinking ability, but also enriches the methods and means to study mathematical problems.
Pythagorean theorem, as an important and famous theorem in geometry, is widely used not only in mathematics, but also in other natural sciences [back]