Class 9, Grade 2: Li Luyang
Proof of Pythagorean Theorem: Among these hundreds of proof methods, some are very wonderful, some are very concise, and some are very famous for their special identities.
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
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.
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''.
Therefore, 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.