In China, the characteristic that the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse is called Pythagorean Theorem or Pythagorean Theorem.
Theorem:
If the two right angles of a right triangle are A and B and the hypotenuse is C, then A 2+B 2 = C 2; That is to say, the sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
If the three sides A, B and C of a triangle satisfy A 2+B 2 = C 2, for example, a right-angled side is 3, a right-angled side is 4, and the hypotenuse is 3×3+4×4=X×X, and X=5. Then this triangle is a right triangle. (called the inverse theorem of Pythagorean theorem)
The origin of Pythagorean theorem;
Pythagoras tree Pythagoras tree is a basic geometric theorem, which is traditionally proved by Pythagoras in ancient Greece. It is said that after Pythagoras proved this theorem, he beheaded a hundred cows to celebrate, so it is also called "Hundred Cows Theorem". In China, the formula and proof of Pythagorean Theorem are recorded in Zhou Kuai Shu Jing, which is said to have been discovered by Shang Dynasty Shang Gao, so it is also called Shang Gao Theorem. During the Three Kingdoms period, Zhao Shuang made a detailed annotation on the Pythagorean Theorem in Zhou Bi suan Jing and gave another proof [5]. France and Belgium are called donkey bridge theorem, and Egypt is called Egyptian triangle. In ancient China, the shorter right-angled side of a right-angled triangle was called a hook, the longer right-angled side was called a chord, and the hypotenuse was called a chord.
Books on Pythagorean Theorem
Principles of Mathematics People's Education Press
Exploring Pythagorean Theorem Tongji University Press
Peking University Press: You Yinpei teaches mathematics.
Pythagoras model new century publishing house
Book nine chapters arithmetic
You Yinpei Reveals Pythagorean Theorem Jiangxi Education Publishing House
pythagoras tree
Pythagoras tree is a graph drawn by Pythagoras according to Pythagoras theorem, which can be repeated indefinitely. Because it is shaped like a tree after repeated several times, it is called Pythagoras tree.
The sum of squares of two right angles of a right triangle is equal to the square of the hypotenuse.
The sum of the areas of two adjacent small squares is equal to the area of one adjacent large square.
Using the inequality A2+B2≥2AB
The area of the triangle between three squares is less than or equal to a quarter of the area of a large square and more than or equal to half of the area of a small square. [Edit this paragraph] The earliest application of Pythagorean theorem It can be seen from many clay tablets that Babylonians were the first people to discover Pythagorean theorem in the world. This is just an example. For example, in BC 1700, the ninth question on a clay tablet (No.BM85 196) was to the effect that "there is a wooden beam (AB) with a length of 5 meters vertically leaning against the wall, and the upper end (A) slides down one meter to d .. How far is the lower end (C) from the wall root (B)?" They solved the problem with Pythagorean theorem, as shown in the figure.
Let AB = CD = L = 5m, BC=a, AD = H = 1m, BD = L-H = 5- 1m = 4m.
∴ A = √ [L-(L-H)] = √ [5-(5-1)] = 3m, ∴ Triangle BDC is a twisted triangle with 3, 4 and 5 sides. [Edit this paragraph] The formula and proof of Pythagorean theorem in Zhou Kuai Suan Jing, one of the ten books in Zhou Kuai Suan Jing. 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.
First of all, the formula of Pythagorean theorem is clearly recorded in Weekly Parallel Calculations: "If you seek evil to the sun, take sunset as a sentence, and the height of the sun as a share. Multiply each sentence and share it separately, and divide it by the prescription to get evil to the sun" (Volume II of Weekly Parallel Calculations)
The proof of Pythagorean Theorem is in Volume I of Weekly Calculations [1]-
The former Duke of Zhou asked Shang Gao, "I heard that doctors are good at counting. Can the old man set a calendar for a week-the sky can rise without steps, but the earth can't be measured? How can it be counted? "
Shang Gao said: "The counting method comes from the square, the circle from the square, the square from the moment, and the moment from 998 1. So I folded it instantly, thinking that the sentence is three in width, four in thigh and five in diameter. If the square is square, the outer half is moment, and the ring is * * *, that is 345. The two moments * * * are twenty and five respectively, which are called product moments. Therefore, the reason why Yu ruled the world was born of this number. "
Duke Zhou felt incredible about the story of the ancient Fuxi (Bao) structure of the weekly calendar (the sky can't rise step by step and the earth can't be measured), so he asked Shang Gao where his mathematical knowledge came from. So quotient Gao takes the proof of Pythagorean theorem as an example to illustrate the origin of mathematical knowledge.
The book Weekly Parallel Computing proves that "the counting method comes from the circle, the circle comes from the square, the square comes from the moment, and the moment comes from 998 1." Explain the development thread-the method of number comes from the circle (π3) square (square), the circle comes from the square (circle area = circumscribed circle *π/4), the square comes from the moment (square comes from the equilateral moment), and the moment comes from 998 1 (the calculation basis of length multiplied by wide area is the multiplication table).
"So the moment is 1, which means that the sentence width is three, the thigh is four and the diameter is five." : Start drawing —— Select a moment with hook three (π three) and strand four (square), and the connecting line on both sides of the moment should be 5 (radius angle five).
"(2) If the square is square, the outer half is moment, and the ring is * * *, it is 345." This is the key proof process-draw a square (hook, square) with two sides of the moment, draw another moment according to the chord of the moment (ruler, actually used as a right triangle), cut the triangle obtained by the "outer half of the moment" and copy it around to form a big square. As you can see, there are three squares, three hooks, four squares and five chords.
"The two moments * * * are 320 and 5, which are called product moments." This is the checking calculation-the sum of the square and the square area is equal to the area of the chord 25-from the graphic point of view, a large square MINUS four triangular areas is the chord, and then a large square MINUS the upper right and lower left rectangular areas is the sum of the square areas. Because a triangle is half the area of a rectangle, it can be deduced that the areas of four triangles are equal to the areas of the upper right and lower left rectangles, so hook+chord = chord.
note:
① Moment, also called square, is an L-shaped woodworking tool, which is a right angle composed of long and short pieces of wood. In ancient times, "moment" refers to an L-shaped ruler, and "moment" is a rectangle derived from "moment".
(2) The sentence "What is right is right, and what is outside is half right" is controversial. The Qing Dynasty edition of Sikuquanshu was defined as "half-carved outside the square", while the previous editions were mostly "half-carved outside the square". Scholars such as Liang-Tso Chen [2], Li Guowei [3], Li Jimin [4] and Qu Anjing [5] have all studied it, and it is more logical to say that "the outside is half an hour".
③ Length refers to area. In ancient times, the dimensional comparison of different dimensions did not invent a new term, but was collectively called "dragon" Zhao Shuang pointed out: "In two moments, every sentence shared the reality." . * * * Older people, actual number.
Due to the long history, the string diagram of Duke Zhou was lost, and the handed down version only printed the string diagram of Zhao Shuang (papermaking was invented in Han Dynasty). Therefore, some scholars mistakenly thought that Shang Gao did not prove it (just said something inexplicable), and later Zhao Shuangcai gave proof.
Actually, it is not. Excerpts from Zhao Shuang's comments on Zhou Kuai Shu Jing [1] when the sentence and the square diagram of the stock-"The sentence and the stock are multiplied separately, and they are combined into a string, and the roots are divided into a string. Case: The string diagram can be multiplied by the sentence stock into Zhu Shi, and then multiplied by Zhu Shi. The difference between sentences and stocks is multiplied into a medium yellow reality, and the difference is also a string reality. "
Zhao Shuang's string diagram stresses that the string diagram is OK, and the string is solid. The words "You" and "Yi" indicate that Zhao Shuang thinks Pythagorean theorem can be proved in another way, so he gives a new proof.
The following is Zhao Shuang's proof-
The triangle in the blue-red picture is a right triangle, the square with hook A as the side is a red square, and the square with rope B as the side is a blue square. Zhu Fang and Fang Qing were combined into a string phalanx. According to its area relationship, there is a 2+b 2 = c 2. Because Zhu Fang and Fang Qing each have a part in metaphysics, that part will not move.
The square with the hook as the edge is Zhu Fang, and the square with the rope as the edge is Fang Qing. To make up for the deficiency, just move Zhu Fang's I(a2) to I', Fang Qing's II to II', and III to III'. In the figure, a square (C ... 2) with the chord as the side length is made up. It can be proved that A+B 2 = C 2; [Edit this paragraph] The story of Garfield proving Pythagorean Theorem 1876 One weekend evening, a middle-aged man was walking in the suburbs of Washington, DC, enjoying the beautiful scenery at dusk. He was Garfield, a senator from Ohio, then a party member. 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 said, "Sir, can you tell me the truth?" Garfield was speechless, unable to explain, and very unhappy. 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.
As follows:
Solution: In the grid, the sum of the areas of two small squares with right angles is equal to the areas of two squares with hypotenuse.
The content of Pythagorean theorem: the sum of squares of two right angles A and B of a right triangle is equal to the square of hypotenuse C,
Square of a+square of b = square of c;
Explanation: Ancient scholars in China called the shorter right-angled side of right-angled triangle "hook", the longer right-angled side "chord" and the hypotenuse "chord", so they called this theorem "Pythagorean Theorem". Pythagorean theorem reveals the relationship between the sides of a right triangle.
For example, the two right-angled sides of a right-angled triangle are 3 and 4 respectively, then the square of the hypotenuse c; = square of a+square of b =9+ 16=25, that is, c=5.
Then the hypotenuse is 5.
[Edit this paragraph] There are many ways to prove Pythagorean theorem, and the method to prove it may be the most among many mathematical theorems. Elisha Scott Loomis's Pythagorean proposition always mentions 367 ways of proof.
Some people will try to prove Pythagorean theorem by trigonometric identities (such as Taylor series of sine and cosine functions), but all basic trigonometric identities are based on Pythagorean theorem, so they cannot be used as proof of Pythagorean theorem (see circular argument).
Proof method 1 (Mei Wending proof)
Make four congruent right-angled triangles, let their two right-angled sides be A and B respectively, and the hypotenuse be C, and make them into polygons as shown in the figure, so that D, E and F are in a straight line. As an extension of AC passing through C, it intersects DF at point P. 。
∫D, e, f are in a straight line, rtδGEF≌rtδEBD,
∴ ∠EGF = ∠BED，
∫∠EGF+∠GEF = 90，
∴ ∠BED + ∠GEF = 90，
∴ ∠BEG = 180 ―90 = 90
AB = BE = EG = GA = c，
Abeg is a square with a side length of C.
∴ ∠ABC + ∠CBE = 90
∫rtδABC≌rtδEBD，
∴ ∠ABC = ∠EBD。
∴ ∠EBD + ∠CBE = 90
That is ∠ CBD = 90.
∠∠BDE = 90，∠ BCP = 90，
BC = BD = a。
BDPC is a square with a side length of 100.
Similarly, HPFG is a square with a side length of B.
Let the area of polygon GHCBE be s, then
,
∴ .
Proof Method 2 (Proof to Minda)
Make two congruent right-angled triangles, and let their two right-angled sides be A and B (B >; A), the hypotenuse length is C. Then make a square with a side length of C, and make them into polygons as shown in the figure, so that E, A and C are in a straight line.
QP∨BC is the point passing through Q, and P is the point passing through AC.
Point B is BM⊥PQ, and the vertical foot is m; A little more.
F is FN⊥PQ, and the vertical foot is n.
∫∠BCA = 90, QP∨ BC.
∴ ∠MPC = 90，
* bm⊥pq，
∴ ∠BMP = 90，
∴ BCPM is a rectangle, that is ∠ MBC = 90.
∠∠QBM+∠MBA =∠QBA =，
∠ABC + ∠MBA = ∠MBC = 90
∴ ∠QBM = ∠ABC，
∵∠ BMP = 90，∠ BCA = 90，BQ = BA = c，
∴rtδbmq≌rtδBCA。
Similarly, rt δ qnf ≌ rt δ AEF can also be proved.
Proof Method 3 (Zhao Haojie Proof)
Make two congruent right-angled triangles, and let their two right-angled sides be A and B (B >; A), the hypotenuse is C. Then make a square with a side length of C and put them together to form a polygon as shown in the figure.
Make square FCJI and AEIG with CF and AE as side lengths respectively,
∫EF = DF-DE = b-a，EI=b，
∴FI=a，
∴ g, I and j are on the same line,
CJ = CF = a，CB=CD=c，
∠CJB = ∠CFD = 90，
∴rtδcjb rtδCFD，
Similarly, rt Δ abg ≌ rt Δ ade,
∴rtδcjb≌rtδCFD≌rtδabg≌rtδade
∴∠ABG = ∠BCJ，
∠∠BCJ+∠CBJ = 90，
∴∠ABG +∠CBJ= 90，
∫∠ABC = 90 °,
∴ g, b, I and j are on the same line,
Proof 4 (Euclid Proof)
Make three squares with sides a, b and c, and put them in the shape as shown in the figure, so that H, c and b are connected into a straight line.
BF，CD。 Exceeding c as CL⊥DE,
AB crosses at m and DE crosses at l.
AF = AC，AB = AD，
∠FAB = ∠GAD，
∴δfab≌δgad，
∫δFAB has an area equal to,
The area of GAD is equal to the right angle ADLM.
Half the area,
The ∴ area of rectangle ADLM =.
Similarly, the area of rectangular MLEB =.
The area of ADEB square
= area of rectangular ADLM+area of rectangular MLEB
∴ That is, the square of A+the square of B = the square of C.
Proof 5 Euclid's Proof
Proof in the Elements of Geometry
In Euclid's Elements of Geometry, the Pythagorean theorem was proved as follows. Let △ABC be a right triangle, where A is a right angle. Draw a straight line from point A to the opposite side so that it is perpendicular to the opposite square. This line divides the opposite square in two, and its area is equal to the other two squares.
In formal proof, we need the following four auxiliary theorems:
If two triangles have two sets of corresponding sides and the angles between the two sets of sides are equal, then the two triangles are congruent. (SAS Theorem) The area of a triangle is half that of any parallelogram with the same base and height. The area of any square is equal to the product of its sides. The area of any square is equal to the product of its two sides (according to Auxiliary Theorem 3). The concept of proof is: transform the upper two squares into two parallelograms with equal areas, and then rotate and transform them into the lower two rectangles with equal areas.
This is proved as follows:
Let △ABC be a right triangle, and its right angle is CAB. Its sides are BC, AB and CA, which are drawn into four squares in turn: CBDE, Baff and ACIH. Draw parallel lines where BD and CE intersect with point A, and this line will intersect BC and DE at right angles at points K and L respectively. Connect CF and AD respectively to form two triangles BCF and BDA. ∠CAB and ∠BAG are right angles, so C, A and G are all linear correspondences, and B, A and H can also be proved in the same way. ∠CBD and∠ ∠FBA are right angles, so∠ ∠ABD is equal to∠ ∠FBC. Because AB and BD are equal to FB and BC respectively, △ABD must be equal to △FBC. Because a corresponds linearly to k and l, the square of BDLK must be twice that of △ABD. Because c, a and g are collinear, the square of BAGF must be twice the area of △FBC. Therefore, the quadrilateral BDLK must have the same area BAGF = AB^2. Similarly, the quadrilateral joint sealant must have the same area ACIH = AC^2. Add these two results, ab^2+ ac^2；; = BD×BK+KL×KC Since BD=KL, BD×BK+KL×KC = BD(BK+KC) = BD×BC Since CBDE is a square, AB 2+AC 2 = BC 22. This proof is the alias Pythagorean Theorem of Pythagorean Theorem proposed by Euclid's Elements of Geometry in section 1.47. It is a dazzling pearl in geometry, known as the "cornerstone of geometry", and it is also widely used in higher mathematics and other disciplines. Because of this, several ancient civilizations in the world have been discovered and widely studied, so there are many names.
China is one of the earliest countries to discover and study Pythagorean theorem. Ancient mathematicians in China called the right triangle pythagorean, the short side of the right angle is called hook, the long side of the right angle is called strand, and the hypotenuse is called chord, so the pythagorean theorem is also called pythagorean chord theorem. BC 1000 years, according to records, Shang Gao (about BC 1 120) replied to the Duke of Zhou, "Therefore, the moment is folded, thinking that the sentence is three in width, four in stock and five in diameter." If the square is square, the outer half is moment, and the ring is * * *, that is 345. Two moments * * * are twenty and five, which are called product moments. Therefore, Pythagorean theorem is also called "quotient height theorem" in China. In the 7th and 6th centuries BC, China scholar Chen Zi once gave the trilateral relationship of any right triangle, that is, "Take the sun as a hook, take the sun as a share, multiply and divide the hook and the share, and evil will come to heaven.
In France and Belgium, Pythagorean Theorem is also called "Donkey Bridge Theorem". Other countries call Pythagorean Theorem "Square Theorem".
One hundred and twenty years after Chen Zi's death, the famous Greek mathematician Pythagoras discovered this theorem, so many countries in the world called it Pythagoras theorem. In order to celebrate the discovery of this theorem, the Pythagorean school killed one hundred cows as a reward for offering sacrifices to the gods, so this theorem is also called the "Hundred Cows Theorem".
Garfield, 20th President of the United States, proved Pythagorean Theorem (1876 April 1).