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Verification of the Pythagorean theorem
There is a very important theorem in trigonometry. Our country calls it the Pythagorean theorem, also known as the Shang-Gao theorem. Because "Zhou Bi Suan Jing" mentioned that Shang Gao said "hook three strands, four strings and five". Several of these proofs are introduced below.
The original proof was of the split type. Let a and b be the right-angled sides of a right triangle, and c be the hypotenuse. Consider the two squares A and B in the figure below with side lengths a+b. Divide A into six parts and B into five parts. Since eight small right triangles are congruent, by subtracting equal quantities from equal quantities, we can deduce that the square on the hypotenuse is equal to the sum of the squares on the two right sides. The quadrilateral in B here is a square with side length c because the sum of the three interior angles of a right triangle is equal to two right angles. The above proof method is called the subtraction congruence proof method. Picture B is the "string picture" in my country's "Zhou Bi Suan Jing".
The picture below is H. The proof given by Perigal in 1873 is an additive congruence proof. In fact, this proof was rediscovered, because this method of division was already known to labitibn Qorra (826~901). (For example: the picture on the right) The following proof was given by H.E. Dudeney in 1917. It also uses a method of proof of additive congruence.
The proof method in the figure below is said to have been designed by Leonardo da Vinci (da Vinci, 1452～1519), and uses the subtraction congruence proof method.
Euclid gave an extremely clever proof of the Pythagorean Theorem in Proposition 47 of Volume 1 of his Elements, as shown in the picture on the next page. Because of the beautiful graphics, some people call it "the monk's turban", while others call it "the bride's sedan chair", which is really interesting. Professor Hua Luogeng once suggested sending this picture to the universe to communicate with "aliens". The outline of the proof is:
(AC)2=2△JAB=2△CAD=ADKL.
Similarly, (BC)2=KEBL
So
(AC)2+(BC)2=ADKL+KEBL=(BC)2 < /p>
The Indian mathematician and astronomer Bhaskara (active around 1150) gave a wonderful proof of the Pythagorean theorem, which is also a segmented proof. As shown in the picture below, divide the square on the hypotenuse into five parts. Four of the parts are triangles that are congruent with the given right triangle; one part is a small square with the difference between the two right-angled sides being the side length. It is easy to put the five pieces back together to get the sum of the squares with two right-angled sides. In fact,
Bashikara also gave a demonstration method as shown below. Draw the height on the hypotenuse of the right triangle to get two pairs of similar triangles, thus we have
c/b=b/m, c/a=a/n, cm=b2 cn=a2 Add both sides Obtain
a2+b2=c(m+n)=c2
This proof was given by the British mathematician J. in the seventeenth century. Rediscovered by Wallis (1616～1703).
Several U.S. presidents have subtle connections with mathematics. G. Washington was once a famous surveyor. T. Jefferson had vigorously promoted higher mathematics education in the United States. A. Lincoln learned logic by studying Euclid's Elements. Even more creative was the seventeenth President J. A. Garfield (1831～1888) had a strong interest and superb talent in elementary mathematics when he was a student. In 1876 (when he was a member of the House of Representatives and five years later elected President of the United States) he gave a beautiful proof of the Pythagorean Theorem, which was published in the New England Journal of Education. The idea of ????the proof is to use the area formulas of trapezoids and right triangles. As shown in the picture on the next page, it is a right-angled trapezoid composed of three right-angled triangles. Use different formulas to find the same area
That is,
a2+2ab+b2=2ab+c2
a2+b2=c2
< p>This kind of proof is often of interest to middle school students when they learn geometry.There are many ingenious ways to prove this theorem (it is said that there are nearly 400 kinds). Here are a few introduced to the students. They are all proved using the puzzle method.
Proof 1 As shown in Figure 26-2, draw squares ABDE, ACFG, and BCHK outside the right triangle ABC. Their areas are c2, b2, and a2 respectively. We only need to prove that the area of ??the big square is equal to the sum of the areas of the two small squares.
Cross C to CM‖BD, cross AB to L, and connect BC and CE.
Because
AB=AE, AC=AG ∠CAE=∠BAG,
So △ACE≌△AGB
And
So
SAEML=SACFG (1)
The same method can be proved
SBLMD=SBKHC (2)
(1)+(2 ) Obtain
SABDE=SACFG+SBKHC,
That is, c2=a2+b2
Proof method 2 As shown in Figure 26-3 (Zhao Junqing’s diagram), use eight The right triangle ABC is assembled into a large square CFGH, whose side length is a+b. Inside it is an inscribed square ABED, whose side length is c, as can be seen from the figure.
SCFGH=SABED+4×SABC,
So a2+b2=c2
Proof 3 is shown in Figure 26-4 (Mei Wending diagram).
Construct a square ABDE outward on the hypotenuse AB of the right angle △ABC, and a square ACGF on the right angle AC. It can be proved (omitted) that extending GF must pass E; extending CG to K so that GK=BC=a, connecting KD, and constructing DH⊥CF in H, then DHCK is a square with side length a. Suppose
the area of ??the pentagon ACKDE = S
On the one hand,
S = the area of ??the square ABDE + 2 times the area of ??△ABC
=c2+ab (1)
On the other hand,
S=area of ??square ACGF+area of ??square DHGK
+2 times the area of ??△ABC
p>=b2+a2+ab. (2)
From (1), (2)
c2=a2+b2
Proof 4 As shown in Figure 26-5 (Xianmingda diagram), draw a square ABDE on the hypotenuse of the right triangle ABC, and then use the two right-angled sides CA and CB of the right triangle ABC as the basis to complete a square with side length b BFGJ (Figure 26-5). It can be proved (omitted) that the extension line of GF must pass through D. Extend AG to K so that GK=a, and let EH⊥GF in H, then EKGH must be a square with side length equal to a.
Suppose the area of ??pentagon EKJBD is S. On the one hand
S=SABDE+2SABC=c2+ab (1)
On the other hand,
S=SBEFG+2?S△ABC+SGHFK
=b2+ab+a2
From (1), (2)
c2=a2+b2
Pythagorean The theorem is one of the theorems with the most proven methods in mathematics - there are more than 400 ways to prove it! But the first recorded proof, Pythagoras' method, has been lost. The earliest proof method currently available belongs to the ancient Greek mathematician Euclid. His proof took the form of deductive reasoning and was recorded in the masterpiece of mathematics "Elements of Geometry". Among ancient Chinese mathematicians, the first person to prove the Pythagorean Theorem was Zhao Shuang, a mathematician from the State of Wu during the Three Kingdoms period. Zhao Shuang created a "Pythagorean Circle and Square Diagram" and gave a detailed proof of the Pythagorean theorem by combining numbers and shapes. In this "Pythagorean Square Diagram", the square ABDE obtained with the chord as the side length is composed of 4 equal right triangles plus the small square in the middle. The area of ??each right triangle is ab/2; the side length of the small square in the middle is b-a, so the area is (b-a) 2. So we can get the following formula: 4×(ab/2)+(b-a) 2 =c 2 After simplification, we can get: a 2 +b 2 =c 2 That is: c= (a 2 +b 2 ) (1/2) Zhao Shuang’s proof is unique and highly innovative. He used the truncation, cutting, spelling and complementing of geometric figures to prove the identity relationship between algebraic expressions, which is both rigorous and intuitive. He provided the basis for ancient China to demonstrate numbers, unify shapes and numbers, and closely combine algebra and geometry. A unique style that is inseparable from each other sets an example. The following URL is Zhao Shuang's "Pythagorean Square Diagram": Most mathematicians in the future inherited this style and developed it, but the division, union, transfer and complementation of the specific figures were slightly different. For example, when Liu Hui proved the Pythagorean theorem later, he also used the method of formal proof of numbers. Liu Hui used the "complementary method of entry and exit", that is, the cut-and-paste proof method. He put certain areas on the square with Pythagorean as the side. Cut it out (out) and move it to the blank area of ??the square with the string as the side (in). The result is that it just fills up. The problem is solved completely using the diagram method. The following URL is Liu Hui's "Green and Zhu Entrance and Exit Picture":
The application of the Pythagorean Theorem is very wide. Another ancient book from the Warring States Period in my country, "Twelve Notes to the Postscript of Lu Shi" contains this record: "Yu controlled the floods and cut off the rivers. He looked at the shapes of the mountains and rivers, determined the high and low trends, eliminated monstrous disasters, and directed the waters into the East China Sea. There is no danger of drowning, so this Pythagorean phenomenon is related to life. "The meaning of this passage is that in order to control floods, Dayu made rivers flow freely, determined the direction of water flow according to the height of the terrain, and took advantage of the situation to make the flood flow into the sea. , there will no longer be flood disasters, which is the result of applying the Pythagorean theorem.
The Pythagorean Theorem has a wide range of applications in our lives.
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