Draw two squares with side length (a+b), as shown in the figure, where A and B are right-angled sides and C is hypotenuse. The two squares are congruent, so the areas are equal.
The left picture and the right picture each have four triangles that are the same as the original right triangle, and the sum of the areas of the left and right triangles must be equal. If all four triangles in the left and right images are deleted, the areas of the rest of the image will be equal. There are two squares left on the left, with A and B as sides respectively. On the right is a square with C as its side. therefore
a2+b2=c2 .
This is the method introduced in our geometry textbook. Intuitive and simple, everyone can understand.
2. The Greek method
Draw squares directly on three sides of a right triangle, as shown in the figure.
It's easy to see,
△ABA '?△AA ' ' C .
Draw a vertical line through C to a' b', cross AB at C' and cross A' b' at C'.
△ ABA ′ and square ACDA'' ′′′′′ have the same base height, the former is half the area of the latter, and the △ AA ′″ c and rectangle AA ′″″ c are the same, and the former is half the area of the latter. From △ ABA '△ AA'' C, we can see that the area of square ACDA' is equal to that of rectangle AA''C''C'. Similarly, the area of square BB'EC is equal to the area of rectangle b'' BC'' C''.
So,
S squared AA''B''B=S squared ACDA'+S squared BB'EC,
That is, a2+b2=c2.
As for the triangle area, it is half of the rectangular area with the same base and height, which can be obtained by digging and filling method (please prove it yourself). Only the simple area relation is used here, and the area formulas of triangles and rectangles are not involved.
This is the proof of the ancient Greek mathematician Euclid in the Elements of Geometry.
The above two proof methods are wonderful because they use few theorems and only use two basic concepts of area:
(1) The area of congruence is equal;
⑵ Divide a graph into several parts, and the sum of the areas of each part is equal to the area of the original graph.
This is a completely acceptable simple concept that anyone can understand.
Mathematicians in China have demonstrated Pythagorean Theorem in many ways, and illustrated Pythagorean Theorem in many ways. Among them, Zhao Shuang (Zhao) proved Pythagorean Theorem in his paper Pythagorean Diagrams, which was attached to Zhou Bi Shu Jing. Use cut and fill method:
As shown in the figure, the four right-angled triangles in the figure are colored with cinnabar, and the small square in the middle is colored with yellow, which is called the middle yellow solid, and the square with the chord as the side is called the chord solid. Then, after patchwork and matching, he affirmed that the relationship between pythagorean chords conforms to pythagorean theorem. That is, "Pythagoras shares multiply each other, and they are real strings, and they are divided, that is, strings."
Zhao Shuang's proof of Pythagorean theorem shows that China mathematicians have superb ideas of proving problems, which are concise and intuitive.
Many western scholars have studied Pythagoras theorem and given many proof methods, among which Pythagoras gave the earliest proof in written records. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate. Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.
The following is the proof of Pythagorean theorem by Garfield, the twentieth president of the United States.
As shown in the figure,
S trapezoid ABCD= (a+b)2
= (a2+2ab+b2)，①
And s trapezoidal ABCD=S△AED+S△EBC+S△CED.
= ab+ ba+ c2
= (2ab+c2).②
Comparing the above two formulas, we can get
a2+b2=c2 .
This proof is quite concise because it uses trapezoidal area formula and triangular area formula.
On April 1876, Garfield published his proof of Pythagorean theorem in the New England Journal of Education. Five years later, Garfield became the twentieth president of the United States. Later, in order to commemorate his intuitive, simple, easy-to-understand and clear proof of Pythagorean theorem, people called this proof "presidential proof" of Pythagorean theorem and it was passed down as a story in the history of mathematics.
After studying similar triangles, we know that in a right triangle, the height on the hypotenuse divides the right triangle into two right triangles similar to the original triangle.
As shown in the figure, in Rt△ABC, ∠ ACB = 90. Make CD⊥BC, while foothold is D.
△BCD∽△BAC，△CAD∽△BAC .
From △BCD∽△BAC, we can get BC2=BD? BA，①
AC2=AD can be obtained from △CAD∽△BAC? AB .②
We found that by adding ① and ②, we can get.
BC2+AC2=AB(AD+BD)，
And AD+BD=AB,
So there is BC2+AC2=AB2, that is
a2+b2=c2 .
This is also a method to prove Pythagorean theorem, and it is also very concise. It makes use of similar triangles's knowledge.
In the numerous proofs of Pythagorean theorem, people also make some mistakes. If someone gives the following methods to prove Pythagorean theorem:
According to the cosine theorem, let △ABC, ∠ c = 90.
c2=a2+b2-2abcosC，
CosC=0 because ∠ c = 90. therefore
a2+b2=c2 .
This seemingly correct and simple proof method actually makes a mistake in the theory of circular proof. The reason is that the proof of cosine theorem comes from Pythagorean theorem.
People are interested in Pythagorean theorem because it can be generalized.
Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".
From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter".
Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.
If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides.
And so on.
appendix
First of all, it briefly introduces Zhou pian Ji Jing.
Zhou Kuai Kuai Jing is one of the ten books of calculation. Written in the second century BC, it was originally named Zhou Jie, which is the oldest astronomical work in China. It mainly expounded the theory of covering the sky and the method of four seasons calendar at that time. In the early Tang Dynasty, it was stipulated as one of imperial academy's teaching materials, so it was renamed Zhou Kuai. The main achievement of Zhouyi ·suan Jing in mathematics is the introduction of Pythagorean theorem and its application in measurement. The original book did not prove Pythagorean theorem, but the proof was given by Zhao Shuang in Zhou Zhuan Pythagorean Notes.
·suan Jing of Zhouyi adopts quite complicated fractional algorithm and Kaiping method.
Second, the story of Garfield proving Pythagorean theorem
1876 One weekend evening, on the outskirts of Washington, D.C., a middle-aged man was walking and enjoying the beautiful scenery in the evening. He was Ohio and party member Garfield. Walking, he suddenly found two children talking about something with rapt attention on a small stone bench nearby, arguing loudly and discussing in a low voice. Driven by curiosity, Garfield followed the sound and came to the two children to find out what they were 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 five." 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 was speechless, unable to explain, and very unhappy.
So Garfield stopped walking and immediately went home to discuss the questions the little boy gave him. After repeated thinking and calculation, he finally figured it out and gave a concise proof method.
Quoted from: /education/yanjiu/ The Discovery of Mathematics. The theorem that graphics cannot be transferred and pasted proves to have unparalleled charm.
-Proof of Pythagorean Theorem
Pythagorean theorem is a pearl in geometry, so it is full of charm. For thousands of years, people have been eager to prove it, including famous mathematicians, amateur mathematicians, ordinary people, distinguished dignitaries and even national presidents. Perhaps it is precisely because of the importance, simplicity and attractiveness of Pythagorean theorem that it has been repeatedly hyped and demonstrated for hundreds of times. 1940 published a proof album of Pythagorean theorem, which collected 367 different proof methods. In fact, that's not all. Some data show that there are more than 500 ways to prove Pythagorean theorem, and only the mathematician Hua in the late Qing Dynasty provided more than 20 wonderful ways to prove it. This is unmatched by any theorem.
Among these hundreds of methods of proof, some are very wonderful, some are very concise, and some are very famous because of the special identity of witnesses.
Firstly, the two most wonderful proofs of Pythagorean theorem are introduced, which are said to come from China and Greece respectively.
1 China method
Draw two squares with side length (a+b), as shown in the figure, where A and B are right-angled sides and C is hypotenuse. The two squares are congruent, so the areas are equal.
The left picture and the right picture each have four triangles that are the same as the original right triangle, and the sum of the areas of the left and right triangles must be equal. If all four triangles in the left and right images are deleted, the areas of the rest of the image will be equal. There are two squares left in the picture on the left, with A and B as sides respectively. On the right is a square with C as its side. therefore
a2+b2=c2 .
This is the method introduced in our geometry textbook. Intuitive and simple, everyone can understand.
2. The Greek method
Draw squares directly on three sides of a right triangle, as shown in the figure.
It's easy to see,
△ABA '?△AA ' ' C .
Draw a vertical line through C to a' b', cross AB at C' and cross A' b' at C'.
△ ABA ′ and square ACDA'' ′′′′′ have the same base height, the former is half the area of the latter, and the △ AA ′″ c and rectangle AA ′″″ c are the same, and the former is half the area of the latter. From △ ABA '△ AA'' C, we can see that the area of square ACDA' is equal to that of rectangle AA''C''C'. Similarly, the area of square BB'EC is equal to the area of rectangle b'' BC'' C''.
So,
S squared AA''B''B=S squared ACDA'+S squared BB'EC,
That is, a2+b2=c2.
As for the triangle area, it is half of the rectangular area with the same base and height, which can be obtained by digging and filling method (please prove it yourself). Only the simple area relation is used here, and the area formulas of triangles and rectangles are not involved.
This is the proof of the ancient Greek mathematician Euclid in the Elements of Geometry.
The above two proof methods are wonderful because they use few theorems and only use two basic concepts of area:
(1) The area of congruence is equal;
⑵ Divide a graph into several parts, and the sum of the areas of each part is equal to the area of the original graph.
This is a completely acceptable simple concept that anyone can understand.
Mathematicians in China have demonstrated Pythagorean Theorem in many ways, and illustrated Pythagorean Theorem in many ways. Among them, Zhao Shuang (Zhao) proved Pythagorean Theorem in his paper Pythagorean Diagrams, which was attached to Zhou Bi Shu Jing. Use cut and fill method:
As shown in the figure, the four right-angled triangles in the figure are colored with cinnabar, and the small square in the middle is colored with yellow, which is called the middle yellow solid, and the square with the chord as the side is called the chord solid. Then, after patchwork and matching, he affirmed that the relationship between pythagorean chords conforms to pythagorean theorem. That is, "Pythagoras shares multiply each other, and they are real strings, and they are divided, that is, strings."
Zhao Shuang's proof of Pythagorean theorem shows that China mathematicians have superb ideas of proving problems, which are concise and intuitive.
Many western scholars have studied Pythagoras theorem and given many proof methods, among which Pythagoras gave the earliest proof in written records. It is said that when he proved Pythagorean theorem, he was ecstatic and killed a hundred cows to celebrate. Therefore, western countries also call Pythagorean Theorem "Hundred Cows Theorem". Unfortunately, Pythagoras' proof method has long been lost, and we have no way of knowing his proof method.
The following is the proof of Pythagorean theorem by Garfield, the twentieth president of the United States.
As shown in the figure,
S trapezoid ABCD= (a+b)2
= (a2+2ab+b2)，①
And s trapezoidal ABCD=S△AED+S△EBC+S△CED.
= ab+ ba+ c2
= (2ab+c2).②
Comparing the above two formulas, we can get
a2+b2=c2 .
This proof is quite concise because it uses trapezoidal area formula and triangular area formula.
On April 1876, Garfield published his proof of Pythagorean theorem in the New England Journal of Education. Five years later, Garfield became the twentieth president of the United States. Later, in order to commemorate his intuitive, simple, easy-to-understand and clear proof of Pythagorean theorem, people called this proof "presidential proof" of Pythagorean theorem and it was passed down as a story in the history of mathematics.
After studying similar triangles, we know that in a right triangle, the height on the hypotenuse divides the right triangle into two right triangles similar to the original triangle.
As shown in the figure, in Rt△ABC, ∠ ACB = 90. Make CD⊥BC, while foothold is D.
△BCD∽△BAC，△CAD∽△BAC .
From △BCD∽△BAC, we can get BC2=BD? BA，①
AC2=AD can be obtained from △CAD∽△BAC? AB .②
We found that by adding ① and ②, we can get.
BC2+AC2=AB(AD+BD)，
And AD+BD=AB,
So there is BC2+AC2=AB2, that is
a2+b2=c2 .
This is also a method to prove Pythagorean theorem, and it is also very concise. It makes use of similar triangles's knowledge.
In the numerous proofs of Pythagorean theorem, people also make some mistakes. If someone gives the following methods to prove Pythagorean theorem:
According to the cosine theorem, let △ABC, ∠ c = 90.
c2=a2+b2-2abcosC，
CosC=0 because ∠ c = 90. therefore
a2+b2=c2 .
This seemingly correct and simple proof method actually makes a mistake in the theory of circular proof. The reason is that the proof of cosine theorem comes from Pythagorean theorem.
People are interested in Pythagorean theorem because it can be generalized.
Euclid gave a generalization theorem of Pythagorean theorem in Elements of Geometry: "A straight side on the hypotenuse of a right triangle has an area equal to the sum of the areas of two similar straight sides on two right angles".
From the above theorem, the following theorem can be deduced: "If a circle is made with three sides of a right-angled triangle as its diameter, the area of the circle with the hypotenuse as its diameter is equal to the sum of the areas of two circles with two right-angled sides as its diameter".
Pythagorean theorem can also be extended to space: if three sides of a right triangle are used as corresponding sides to make a similar polyhedron, then the surface area of a polyhedron on the hypotenuse is equal to the sum of the surface areas of two polyhedrons on the right side.
If three sides of a right-angled triangle are used as balls, the surface area of the ball on the hypotenuse is equal to the sum of the surface areas of two balls made on two right-angled sides.
And so on.
appendix
First of all, it briefly introduces Zhou pian Ji Jing.
Zhou Kuai Kuai Jing is one of the ten books of calculation. Written in the second century BC, it was originally named Zhou Jie, which is the oldest astronomical work in China. It mainly expounded the theory of covering the sky and the method of four seasons calendar at that time. In the early Tang Dynasty, it was stipulated as one of imperial academy's teaching materials, so it was renamed Zhou Kuai. The main achievement of Zhouyi ·suan Jing in mathematics is the introduction of Pythagorean theorem and its application in measurement. The original book did not prove Pythagorean theorem, but the proof was given by Zhao Shuang in Zhou Zhuan Pythagorean Notes.
·suan Jing of Zhouyi adopts quite complicated fractional algorithm and Kaiping method.
Second, the story of Garfield proving Pythagorean theorem
1876 One weekend evening, on the outskirts of Washington, D.C., a middle-aged man was walking and enjoying the beautiful scenery in the evening. He was Ohio and party member Garfield. Walking, he suddenly found two children talking about something with rapt attention on a small stone bench nearby, arguing loudly and discussing in a low voice. Driven by curiosity, Garfield followed the sound and came to the two children to find out what they were 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 five." 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 was speechless, unable to explain, and very unhappy.
So Garfield stopped walking and immediately went home to discuss the questions the little boy gave him. After repeated thinking and calculation, he finally figured it out and gave a concise proof method.
In foreign countries, especially in the west, Pythagorean theorem is usually called Pythagorean theorem. This is because they think that the right triangle has the property of "hook 2+ chord 2= chord 2", and Pythagoras, an ancient Greek mathematician, was the first to give a strict proof.
In fact, in earlier human activities, people have realized some special cases of this theorem. In addition to the Pythagorean theorem discovered in China more than 0/000 years ago, it is said that the ancient Egyptians also used the law of "hooking three strands, four chords and five" to determine the right angle. However, this legend has aroused the suspicion of many mathematical historians. For example, Professor M. Klein, an American mathematical historian, once pointed out: "We don't know whether the Egyptians realized the Pythagorean theorem. We know that they have people who pull the rope (surveyors), but the theory that they tied a knot on the rope, divided the whole length into three sections, 3, 4 and 5, and then used it to form a right triangle has never been confirmed in any literature. " However, archaeologists discovered several pieces of ancient Babylonian clay tablets, which were completed around 2000 BC. According to expert research, one of them is engraved with the following question: "A stick with a length of 30 units stands upright on the wall. How far is its lower end from the corner when its upper end slides down by 6 units? " This is a special case of a triangle with a side length of 3:4:5; Experts also found that there is a strange number table engraved on another board, in which * * * is engraved with four columns and fifteen rows of numbers, which is a Pythagorean number table: the rightmost column is the serial number from 1 to 15, while the left three columns are the values of stocks and hook chords respectively, and a * * * records/kloc-0.
Whether it is the ancient Egyptians, Babylonians or China who first discovered Pythagorean theorem, the same property discovered by our ancestors at different times and places is obviously not only the private property of any nation, but the wealth of all mankind. It is worth mentioning that the gains after discovering the same attribute of this * * * are not exactly the same. The following is based on Pythagorean Theorem and Pythagorean Theorem.
1. Pythagoras theorem
Pythagoras is an ancient Greek name. Pythagoras was born in the 6th century BC. He traveled to Egypt, Babylon (another way of saying it is India) and other places in his early years, and later moved to Crotone in the south of the Italian peninsula, where he organized a secret group-Pythagoras School, which attached great importance to mathematics and tried to explain everything with numbers. They claim that numbers are the origin of everything in the universe. The purpose of learning mathematics is not to be practical, but to explore the mysteries of nature. Their great contribution to mathematics is to consciously admit and emphasize that mathematical things such as figures and numbers are abstractions of thinking, which are completely different from actual things or actual images. Some people in primitive civilized society (such as Egyptians and Babylonians) also know how to think without numbers, but their conscious awareness of the abstraction of this kind of thinking is quite different from that of Pythagoras school. Moreover, before the Greeks, geometric thought was inseparable from physical objects. For example, the Egyptians thought that a straight line was the edge of a tight rope or a field; A rectangle is the boundary of a field. Another feature of the Pythagorean school is that arithmetic and geometry are closely linked.
Because of this, the Pythagorean school found a formula to express the side length of a right triangle with three integers, which belongs to both arithmetic and geometry: if 2n+ 1 and 2n2+2n are two right-angled sides, then the hypotenuse is 2n2+2n+ 1 (although this rule cannot represent all integer Pythagorean arrays). It is for the above reasons that this school. It is found that the so-called "incommensurable measure", such as the ratio of the hypotenuse to the right side of an isosceles right triangle, that is, the ratio of the diagonal of a square to its side, cannot be expressed by the ratio of integers. Because of this, they call those ratios that can be expressed by the ratio of integers "commensurability ratio", which means that two quantities can be measured by commensurability units. A ratio that cannot be expressed in this way is called an incommensurable ratio. As we wrote today, the ratio of 1 is an incommensurable ratio. As for the proof of incommensurability with 1, it is also given by Pythagoras school. This proof points out that if the hypotenuse of an isosceles right triangle can be commensurable with a right angle, then the same number will be both odd and even. The proof process is as follows: Set an isosceles right triangle. Let this ratio be expressed as the ratio of the smallest integer. According to Pythagoras theorem 2=2+2, there is 2 = 22. Since 22 is even, that is, x2 is even, it must be even, because the square of any odd number must be odd (any odd number can be expressed as 2n+ 1) 2 = 4N2+. Inevitably, it is not even but odd. Since it is an even number, it can be set to = 2. So 2 = 42=22. Therefore, 2 = 22. Therefore, 2 is even, so it is even, but at the same time it is odd, which creates a contradiction.
Regarding the proof of Pythagoras theorem, the earliest written material preserved by human beings is proposition 47 in the first volume of Geometry written by Euclid (about 300 BC): "The square on the hypotenuse of a right triangle is equal to the sum of the squares on two right angles". In fact, the Pythagorean school is more concerned with the study of mathematical problems themselves; Ancient Greek mathematics, represented by Pythagoras School, takes spatial form as the main research object and logical deductive reasoning as the main theoretical form. The discovery of Pythagorean theorem (the research and discussion of commensurability ratio and incommensurability ratio) actually led to the discovery of irrational numbers. Although the Pythagorean school was unwilling to accept such numbers, which led to the so-called first mathematical crisis in the history of mathematics, the Pythagorean school's exploration is still indispensable.
Second, China's Pythagorean Theorem
In our country, the earliest record of Pythagoras theorem that can be found so far is "Weekly Parallel Calculations", which was written around 1 century BC, and there was a conversation about 1 1,000 years ago: "The Duke of Zhou asked Shang Gao: I heard that doctors were good at counting, so I wanted to ask the ancients to make weekly calendars and calendars, and my husband and land could not be promoted step by step." Shang Gao said: the counting method comes from the circle. The circle comes from a square, the square comes from an instant, and the instant comes from 99.8 1. So instantly fold, thinking that the hook is three, the strand is four, and the diameter is five. "
There is also a record of Chen Zi's measurement of the sun in The Classic of Parallel Calculation: according to Pythagoras' theorem, Zhou Zi can measure the height and distance of the sun. For example, when calculating the height of the sun and measuring the distance from the surveyor's position to a point below the sun, the method of calculating the distance from the sun is: "If you seek evil from the sun, take the sun as a hook, the height of the sun as a strand, and Pythagoras multiplies it by himself, except it."
Zhoupian Jing is China's earliest mathematical work. This paper mainly talks about the method of learning mathematics, how to use Pythagorean theorem to calculate abstruse distance and complex fraction calculation. In the Tang Dynasty, Zhoupian Jing was designated as a textbook by imperial academy Mathematics Museum, together with nine other mathematical works that appeared in China for more than 1000 years in Han and Tang Dynasties. Later generations generally call these ten books "Ten Classics of Computation", which fully reflect China's mathematical achievements from pre-Qin to early Tang Dynasty. Many of them involve the content of Pythagorean Theorem, especially the theory of right triangle in Chapter 9 of Nine Arithmetic (one of the ten classic books on calculation). The main content of the discussion is Pythagorean Theorem and its application. 22 technologies are put forward. Among them, the sixth question is the famous "leading water to shore": "Today there is a pool, one foot high, and the water grows in its center. As soon as it came out, it rowed the water to the shore and asked about the geometry of water depth and length. " This is a widely circulated topic, and similar topics have appeared again and again in other books, such as Cheng.
According to Pythagorean theorem, our ancestors also invented a measuring torque consisting of a hook ruler and a handle ruler which are perpendicular to each other. For example, The Bian Jing of Zhou Dynasty records the discussion on how to use the moment in business: "The flat moment is straight, the moment is high, the complex moment is deep, and the lying moment knows the distance." Another example is Liu Hui, an outstanding mathematician in the Wei and Jin Dynasties in China. The fourth question in his masterpiece "Archipelago Calculation" is: "Today, the valley is deep, and the moment is flat on the shore, so that the hook is six feet high, and the bottom of the hook is nine feet and one inch, and the moment is set at the top, and the moment is three feet apart, and the bottom of the hook is eight feet and five inches. The geometry of the valley depth is asked."
At present, China's earliest Pythagorean theorem proved to be Zhao Shuang's annotation of Zhouyi Shu Jing in Han Dynasty.
Mathematics in ancient China is different from that in ancient Greece. In fact, the main research object of mathematics in China is not spatial form, but quantitative relationship. Its theoretical form is not a logical deduction system, but an algorithm system centered on problem solving. Different from the way of thinking of ancient Greek mathematics, the way of thinking of ancient mathematicians in China was mainly intuitive thinking, and analogy was the main means to discover and infer results.
For Pythagorean Theorem, the mathematicians in ancient China did not focus only on the strict proof of logical reasoning, nor did they make a fuss about what the incommensurable measure was, but based on an in-depth study of an algorithm that can be used to solve practical problems. By discussing the propositions related to Pythagorean theorem and similar right triangle property theorems within the scope of right triangle, they introduced a combination ratio algorithm-Pythagorean algorithm. Pythagoras takes the concept of similar right-angled triangles as the basic concept and the properties of similar right-angled triangles as the basic properties, so the similarity ratio between similar right-angled triangles constitutes the core of Pythagoras. Pythagoras expressed the principle that the corresponding sides of Pythagoras are proportional, and solved the problem of Pythagoras integer and Pythagoras two-capacity (including circle and square). The theoretical basis of Pythagoras measurement is established. Later, Liu Hui actually defined the theory similar to Pythagorean shape as Pythagorean ratio theory, and clearly put forward the principle of "no loss in cost ratio". He also combined this principle with the proportional algorithm to demonstrate various pythagorean measurement principles, thus establishing a solid theoretical foundation for pythagorean observation in ancient China.
Some experts also pointed out that Pythagorean theorem played a very important role in ancient mathematics in China, and for thousands of years, Chinese geometry with Pythagorean theorem and its application as its core was gradually formed.