What are the methods to prove the Pythagorean theorem?

Proof 2 can be considered a very straightforward proof. The most interesting thing is that if we flip the right triangle in the diagram and put it together in Figure 3 below, we can still use a similar method to prove the Pythagorean theorem.

The most important method is the equal area method, which is to use the change of the graph to prove;

Similar triangles can also be used to prove the Pythagorean theorem.

Pythagorean theoremIn China, the sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse is called the Pythagorean theorem or the Pythagorean theorem, also known as the Pythagorean theorem or Pythagoras theorem. In mathematical formulas, it is often written as a 2 + b 2 = c 2

Prove the Pythagorean theorem.

The 16 methods are as follows:

1. Proof 1 (Zou Yuanzhi certificate);

2. Proofing 2 (textbook proof);

3. Proof 3 (Zhao Shuang's string diagram proof;

4. Evidence 4 (** proof);

5. Evidence 5 (Mei Wending proves that he is congratulating);

6. Evidence 6 (Xiang Mingda Certificate;

7. Evidence VII (Euclid.

proof); 8. Evidence 8 (similar triangles.

Proof of Nature);

9. Evidence 9 (Yang Zuomei certificate);

10, the law ten (Li Rui.

proof); 11. Proofing 11 (proving by the cutting line theorem);

12. Proofing method 12 (proved by using Doremia's theorem);

13. Testimonial-liquid matching method 12 (proved by Doremia's theorem);

14. Evidence 14 (Zhiqin uses counter-evidence to prove);

15. Evidence 15 (Simbson proof);

16. Evidence 16 (Chen Jie's proof).

This is a very common method of proof, which uses area to prove. Take the three sides of the triangle and make three squares, and find that the sum of the areas of the two smaller squares is equal to the larger triangle. The Pythagorean theorem was proven.

Zhao Shuang's chord diagram refers to the formation of a square with four hypotenuse triangles with a long c, a longer right-angled side and a shorter right-angled side c. There is also a smaller square within this larger square. The Pythagorean theorem is calculated by calculating the area of the whole.

The trapezoidal proof method is also a good proof method. That is, choose two identical right-angled triangles, one horizontally and one vertically, to connect the two points at the height. Calculate that the area of the trapezoid is equal to the addition of the areas of the three triangles respectively, thus proving the Pythagorean theorem.

The most basic theorem we use when learning mathematics is the Pythagorean theorem, so what is its proof method? Let's find out.

Euclid proof

The most common way to prove the Pythagorean theorem is Euclid's proof, where the triangle abc is a straight triangle where A is a right angle. Draw a straight line from point A to the opposite edge so that it is perpendicular to the opposite edge. Extending this line divides the square on the opposite side in two, with an area equal to the other two squares.

The following proof of the Pythagorean theorem is given in Euclid's Primitives of Geometry. Let the triangle ABC be a straight triangle, where A is a right angle. Draw a straight line from point A to the opposite edge so that it is perpendicular to the opposite edge.

Extending this line divides the square on the opposite side in two, with an area equal to the other two squares.

Auxiliary theorem

1. If two triangles have two sets of corresponding sides and the angles between these two sets of sides are equal, then the two triangles are congruent.

2. The area of the triangle is half of the area of the parallelogram with the same base and height.

3. The area of any square is equal to the product of its two sides.

4. The area of any rectangle is equal to the product of its two sides.

To sum up, the most common method of proving the Pythagorean theorem is Euclidean proof, and then there are some auxiliary theorem proofs, such as the area of a triangle is half of the area of any parallelogram with the same height at the same base, the area of any square is equal to the product of its two sides, the area of any rectangle is equal to the product of its two sides, etc.

The Pythagorean theorem was also the first indefinite equation in history to give a complete solution, and led to Fermat's theorem. At present, there are about 500 proofs of the Pythagorean theorem, which is one of the most provable theorems among mathematical theorems. The Pythagorean theorem is the most important theorem in plane geometry!

It was the first theorem in history to link numbers with shapes, which opened the beginning of argumentative geometry and even triggered the first mathematical crisis.

There are Zhao Shuangxian's diagram method, Pythagorean method, Chong Shuyan return to the original proof method, the use of triangle similarity derivation and other proof methods, I hope mine is helpful to you.

So far, there are more than 400 methods to prove the Pythagorean definite perturbation lithology, including geometric proof, algebraic proof, dynamic proof, quaternion proof, etc.

The Pythagorean theorem, is a basic geometric theorem that states that the sum of the squares of the two right-angled sides of a right-angled triangle is equal to the square of the hypotenuse. In ancient China, the right triangle was called the Pythagorean shape, and the smaller of the right-angled sides was the hook, and the other long right-angled side was the strand, and the hypotenuse was the string, so this theorem was called the Pythagorean theorem.

The Pythagorean theorem now has about 500 ways to prove it, making it one of the most provable theorems in mathematics. The Pythagorean theorem is one of the important mathematical theorems discovered and proven by mankind in the early days, one of the most important tools for solving geometric problems with algebraic ideas, and one of the links between numbers and shapes.

In China, Shang Gao during the Zhou Dynasty proposed a special case of the Pythagorean theorem of "Pythagorean three, four strings, five". In the West, the Pythagoreans of ancient Greece in the 6th century BC were the first to propose and prove this theorem, and they used the deductive method to prove that the square of the hypotenuse of a right triangle is equal to the sum of the squares of two right angles.

In the 11th century B.C., the mathematician Shang Gao (a native of the early years of the Western Zhou Dynasty) proposed "Gou."

III. Shares. Fourth, string five". Written before the first century B.C., the Zhou Sutra records a conversation between Shang Gao and the Duke of Zhou.

Shang Gao said: "....Therefore, the folding moment, the hook wide three, the stock repair four, the meridian five. Meaning:

When the two right-angled sides of the right-angled triangular finch are 3 (hook) and 4 (strand) respectively, the radial corner (chord) is 5. Later, people simply said that this fact was "Pythagorean four strings five", and according to this allusion, the Pythagorean theorem was called the quotient high theorem.

In the third century AD, Zhao Shuang of the Three Kingdoms period made a detailed annotation of the Pythagorean theorem in the "Zhou Ji Sutra", which was recorded in the "Nine Chapters of Arithmetic" "Pythagorean multiplication, and divide by the square, that is, the string", Zhao Shuang created a "Pythagorean square diagram", and used the combination of numbers and shapes to obtain the method, and gave a detailed proof of the Pythagorean theorem. Later, Liu Hui also proved the Pythagorean theorem in Liu Hui's note.

In the last years of the Qing Dynasty in China, the mathematician Hua Yufang proposed more than 20 methods for the Pythagorean theorem.