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. So what?
a^2+b^2=c^2。 ?
This is the method introduced in our geometry textbook. Intuitive and simple, everyone can understand. ?
2. Greek method: draw squares directly on three sides of a right triangle, as shown in the figure. ?
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,?
Namely. 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:?
⑴? The areas of congruent shapes are equal; ?
⑵? A figure is divided into several parts, and the sum of the areas of each part is equal to the area of the original figure. ?
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 excavation and filling 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. ?
Up there? The last picture is the spelling to prove Pythagorean theorem.
I hope I can help you.