Analysis:
Theorem proof with incomparable 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.
Quote from:. Ntu.edu/education/yanjiu/'s column "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.
Quote from:. Ntu.edu/education/yanjiu/'s column "Discovery of Mathematics". Pictures cannot be reposted. Please check the original text.