Various ways proof of the Pythagorean theorem

student of 9th "A" class

Municipal educational institution secondary school No. 8

Scientific supervisor:

math teacher,

Municipal educational institution secondary school No. 8

Art. Novorozhdestvenskaya

Krasnodar region.

Art. Novorozhdestvenskaya

ANNOTATION.

The Pythagorean theorem is rightfully considered the most important in the course of geometry and deserves close attention. It is the basis for solving many geometric problems, the basis for studying theoretical and practical geometry courses in the future. The theorem is surrounded by a wealth of historical material related to its appearance and methods of proof. Studying the history of the development of geometry instills a love for this subject, promotes the development of cognitive interest, general culture and creativity, and also develops research skills.

As a result of the search activity, the goal of the work was achieved, which was to replenish and generalize knowledge on the proof of the Pythagorean theorem. It was possible to find and consider various methods of proof and deepen knowledge on the topic, going beyond the pages of the school textbook.

The collected material further convinces us that the Pythagorean theorem is a great theorem of geometry and has enormous theoretical and practical significance.

Introduction. Historical background 5 Main part 8

3. Conclusion 19

4. Literature used 20
1. INTRODUCTION. HISTORICAL BACKGROUND.

The essence of the truth is that it is for us forever,

When at least once in her insight we see the light,

And the Pythagorean theorem after so many years

For us, as for him, it is undeniable, impeccable.

To rejoice, Pythagoras made a vow to the gods:

For touching infinite wisdom,

He slaughtered a hundred bulls, thanks to the eternal ones;

He offered prayers and praises after the victim.

Since then, when the bulls smell it, they push,

That the trail again leads people to a new truth,

They roar furiously, so there’s no point in listening,

Such Pythagoras instilled terror in them forever.

Bulls, powerless to resist the new truth,

What remains? - Just closing your eyes, roaring, trembling.

It is not known how Pythagoras proved his theorem. What is certain is that he discovered it under the strong influence of Egyptian science. A special case of the Pythagorean theorem - the properties of a triangle with sides 3, 4 and 5 - was known to the builders of the pyramids long before the birth of Pythagoras, and he himself studied with Egyptian priests for more than 20 years. A legend has been preserved that says that, having proven his famous theorem, Pythagoras sacrificed a bull to the gods, and according to other sources, even 100 bulls. This, however, contradicts information about the moral and religious views of Pythagoras. In literary sources you can read that he “forbade even killing animals, much less feeding on them, for animals have souls, just like us.” Pythagoras ate only honey, bread, vegetables and occasionally fish. In connection with all this, the following entry can be considered more plausible: “... and even when he discovered that in a right triangle the hypotenuse corresponds to the legs, he sacrificed a bull made of wheat dough.”

The popularity of the Pythagorean theorem is so great that its proofs are found even in fiction, for example, in the story “Young Archimedes” by the famous English writer Huxley. The same Proof, but for the special case of an isosceles right triangle, is given in Plato’s dialogue “Meno”.

Fairy tale "Home".

“Far, far away, where even planes don’t fly, is the country of Geometry. In this unusual country there was one amazing city - the city of Teorem. One day I came to this city beautiful girl named Hypotenuse. She tried to rent a room, but no matter where she applied, she was turned down. Finally she approached the rickety house and knocked. A man who called himself Right Angle opened the door to her, and he invited Hypotenuse to live with him. The hypotenuse remained in the house in which Right Angle and his two young sons, named Katetes, lived. Since then, life in the Right Angle house has changed in a new way. The hypotenuse planted flowers on the window and planted red roses in the front garden. The house took the shape of a right triangle. Both legs really liked the Hypotenuse and asked her to stay forever in their house. In the evenings, this friendly family gathers at the family table. Sometimes Right Angle plays hide and seek with his kids. Most often he has to look, and the Hypotenuse hides so skillfully that it can be very difficult to find. One day, while playing, Right Angle noticed an interesting property: if he manages to find the legs, then finding the Hypotenuse is not difficult. So the Right Angle uses this pattern, I must say, very successfully. The Pythagorean theorem is based on the property of this right triangle.”

(From the book by A. Okunev “Thank you for the lesson, children”).

A humorous formulation of the theorem:

If we are given a triangle

And, moreover, with a right angle,

That is the square of the hypotenuse

We can always easily find:

We square the legs,

We find the sum of powers -

And in such a simple way

We will come to the result.

While studying algebra and the beginnings of analysis and geometry in the 10th grade, I became convinced that in addition to the method of proving the Pythagorean theorem discussed in the 8th grade, there are other methods of proof. I present them for your consideration.
2. MAIN PART.

Theorem. In a right triangle there is a square

The hypotenuse is equal to the sum of the squares of the legs.

1 METHOD.

Using the properties of the areas of polygons, we will establish a remarkable relationship between the hypotenuse and the legs of a right triangle.

Proof.

a, c and hypotenuse With(Fig. 1, a).

Let's prove that c²=a²+b².

Proof.

Let's complete the triangle to a square with side a + b as shown in Fig. 1, b. The area S of this square is (a + b)². On the other hand, this square is made up of four equal right-angled triangles, each of which has an area of ​​½ aw  , and a square with side With, therefore S = 4 * ½ aw + c² = 2aw + c².

Thus,

(a + b)² = 2 aw + c²,

c²=a²+b².

The theorem has been proven.
2 METHOD.

After studying the topic “Similar triangles”, I found out that you can apply the similarity of triangles to the proof of the Pythagorean theorem. Namely, I used the statement that the leg of a right triangle is the mean proportional to the hypotenuse and the segment of the hypotenuse enclosed between the leg and the altitude drawn from the vertex of the right angle.

Consider a right triangle with right angle C, CD – height (Fig. 2). Let's prove that AC² +NE² = AB² .

Proof.

Based on the statement about the leg of a right triangle:

AC = , SV = .

Let us square and add the resulting equalities:

AC² = AB * AD, CB² = AB * DB;

AC² + CB² = AB * (AD + DB), where AD+DB=AB, then

AC² + CB² = AB * AB,

AC² + CB² = AB².

The proof is complete.
3 METHOD.

To prove the Pythagorean theorem, you can apply the definition of the cosine of an acute angle of a right triangle. Let's look at Fig. 3.

Proof:

Let ABC be a given right triangle with right angle C. Let us draw the altitude CD from the vertex of right angle C.

By definition of cosine of an angle:

cos A = AD/AC = AC/AB. Hence AB * AD = AC²

Likewise,

cos B = ВD/ВС = ВС/АВ.

Hence AB * BD = BC².

Adding the resulting equalities term by term and noting that AD + DB = AB, we obtain:

AC² + sun² = AB (AD + DB) = AB²

The proof is complete.
4 METHOD.

Having studied the topic “Relationships between the sides and angles of a right triangle”, I think that the Pythagorean theorem can be proven in another way.

Consider a right triangle with legs a, c and hypotenuse With. (Fig. 4).

Let's prove that c²=a²+b².

Proof.

sin B= high quality ; cos B= a/c , then, squaring the resulting equalities, we get:

sin² B= in²/s²; cos² IN= a²/c².

Adding them up, we get:

sin² IN+cos² B=в²/с²+ а²/с², where sin² IN+cos² B=1,

1= (в²+ а²) / с², therefore,

c²= a² + b².

The proof is complete.

5 METHOD.

This proof is based on cutting squares built on the legs (Fig. 5) and placing the resulting parts on a square built on the hypotenuse.

6 METHOD.

For proof on the side Sun we are building BCD ABC(Fig. 6). We know that the areas of similar figures are related as the squares of their similar linear dimensions:

Subtracting the second from the first equality, we get

c2 = a2 + b2.

The proof is complete.

7 METHOD.

Given(Fig. 7):

ABC,= 90° , sun= a, AC=b, AB = c.

Prove:c2 = a2 +b2.

Proof.

Let the leg b A. Let's continue the segment NE per point IN and build a triangle BMD so that the points M And A lay on one side of the straight line CD and, in addition, BD =b, BDM= 90°, DM= a, then BMD= ABC on two sides and the angle between them. Points A and M connect with segments AM. We have M.D. CD And A.C. CD, that means it's straight AC parallel to the line M.D. Because M.D.< АС, then straight CD And A.M. not parallel. Therefore, AMDC- rectangular trapezoid.

In right triangles ABC and BMD 1 + 2 = 90° and 3 + 4 = 90°, but since = =, then 3 + 2 = 90°; Then AVM=180° - 90° = 90°. It turned out that the trapezoid AMDC is divided into three non-overlapping right triangles, then by the area axioms

(a+b)(a+b)

Dividing all terms of the inequality by , we get

Ab + c2 + ab = (a +b) , 2 ab+ c2 = a2+ 2ab+ b2,

c2 = a2 + b2.

The proof is complete.

8 METHOD.

This method is based on the hypotenuse and legs of a right triangle ABC. He constructs the corresponding squares and proves that the square built on the hypotenuse is equal to the sum of the squares built on the legs (Fig. 8).

Proof.

1) DBC= FBA= 90°;

DBC+ ABC= FBA+ ABC, Means, FBC = DBA.

Thus, FBC=ABD(on two sides and the angle between them).

2) , where AL DE, since BD is a common base, DL- total height.

3) , since FB is a foundation, AB- total height.

4)

5) Similarly, it can be proven that

6) Adding term by term, we get:

, BC2 = AB2 + AC2 . The proof is complete.

9 METHOD.

Proof.

1) Let ABDE- a square (Fig. 9), the side of which is equal to the hypotenuse of a right triangle ABC= s, BC = a, AC =b).

2) Let DK B.C. And DK = sun, since 1 + 2 = 90° (as sharp corners right triangle), 3 + 2 = 90° (like the angle of a square), AB= BD(sides of the square).

Means, ABC= BDK(by hypotenuse and acute angle).

3) Let EL D.K., A.M. E.L. It can be easily proven that ABC = BDK = DEL = EAM (with legs A And b). Then KS= CM= M.L.= L.K.= A -b.

4) SKB = 4S+SKLMC= 2ab+ (a - b),With2 = 2ab + a2 - 2ab + b2,c2 = a2 + b2.

The proof is complete.

10 METHOD.

The proof can be carried out on a figure jokingly called “Pythagorean pants” (Fig. 10). Its idea is to transform squares built on the sides into equal triangles that together make up the square of the hypotenuse.

ABC move it as shown by the arrow, and it takes position KDN. The rest of the figure AKDCB equal area of ​​the square AKDC this is a parallelogram AKNB.

A parallelogram model has been made AKNB. We rearrange the parallelogram as sketched in the contents of the work. To show the transformation of a parallelogram into an equal-area triangle, in front of the students, we cut off a triangle on the model and move it down. Thus, the area of ​​the square AKDC turned out to be equal to the area of ​​the rectangle. Similarly, we convert the area of ​​a square into the area of ​​a rectangle.

Introduction

It is difficult to find a person who does not associate the name of Pythagoras with his theorem. Perhaps, even those who have said goodbye to mathematics forever in their lives retain memories of “Pythagorean pants” - a square on the hypotenuse, equal in size to two squares on the sides.

The reason for the popularity of the Pythagorean theorem is triune: it

simplicity - beauty - significance. Indeed, the Pythagorean theorem is simple, but not obvious. This is a combination of two contradictory

began to give her a special attractive force, makes her beautiful.

In addition, the Pythagorean theorem is of great importance: it is used in geometry literally at every step, and the fact that there are about 500 different proofs of this theorem (geometric, algebraic, mechanical, etc.) testifies to the gigantic number of its specific implementations .

In modern textbooks, the theorem is formulated as follows: “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.”

In the time of Pythagoras, it sounded like this: “Prove that a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs” or “The area of ​​a square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on its legs.”

Goals and objectives

The main goal of the work was to showthe importance of the Pythagorean theorem in the development of science and technology of manycountries and peoples of the world, as well as in the most simple and interestingform to teach the content of the theorem.

The main method used in the work wasis a method of organizing and processing data.

Attracting information Technology, diversezili material with various colorful illustrations.

"GOLDEN VERSES" OF PYTHAGORUS

Be fair both in your words and in your actions... Pythagoras (c. 570-c. 500 BC)

Ancient Greek philosopher and mathematiciandeveloped with his teaching about cosmic harmony andtransmigration of souls. Tradition credits Pythagoras with proving the theorem that bears his name. Much inPlato's teachings go back to Pythagoras and his successors tel.

There are no written documents left about Pythagoras of Samos, the son of Mnesarchus, and from later evidence it is difficult to reconstruct the true picture of his life and achievements.(Electronic encyclopedia:StarWorld) It is known that Pythagoras left his native island of Samos in the Aegean Sea at the shoregov of Asia Minor in protest against the tyranny of the ruler and already in adulthoodage (according to legend, 40 years old) appeared in the Greek city of Crotone in southern Italy. Pythagoras and his followers - the Pythagoreans - formed a secret alliance that played a significant role in the life of the Greek colonies in ItaLii. The Pythagoreans recognized each other by a star-shaped pentagon - a pentagram. But Pythagoras had to retire to Metapontus, where hedied. Later in the second halfVBC e., his order was destroyed.

The teachings of Pythagoras were greatly influenced by philosophy and religiongia of the East. He traveled a lot in the countries of the East: he was inEgypt and Babylon. There Pythagoras also met Eastern mathematics tikoy.

The Pythagoreans believed that secrets were hidden in numerical patterns.on the world. The world of numbers lived a special life for the Pythagorean; numbers hadits own special life meaning. Numbers equal to the sum of their divisors were perceived as perfect (6, 28, 496, 8128); friendlynamed pairs of numbers, each of which was equal to the sum of the other's divisorsgogo (for example, 220 and 284). Pythagoras was the first to divide numbers into even andodd, prime and composite, introduced the concept of figured numbers. In hisThe school examined in detail Pythagorean triplets of natural numbers, in which the square of one was equal to the sum of the squares of the other two (Fermat's last theorem).

Pythagoras is credited with saying: “Everything is a number.” To the numbers(and he only meant natural numbers) he wanted to bring the whole world together, andmathematics in particular. But in the school of Pythagoras itself a discovery was made that violated this harmony. It has been proven that the root of 2 is notis a rational number, i.e. it cannot be expressed in terms of natural numbers numbers.

Naturally, Pythagoras’ geometry was subordinated to arithmetic.This was clearly manifested in the theorem that bears his name and later becamethe basis for the application of numerical methods of geometry. (Later, Euclid again brought geometry to the forefront, subordinating algebra to it.) Apparently, the Pythagoreans knew the correct solids: tetrahedron, cube and dodecahedron.

Pythagoras is credited with the systematic introduction of proofs into geometry, the creation of planimetry of rectilinear figures, the doctrine of bii.

The name of Pythagoras is associated with the doctrine of arithmetic, geometric and harmonic proportions.

It should be noted that Pythagoras considered the Earth to be a ball movingaround the sun. When inXVIcentury the church began to be fiercely persecutedIf we take the teaching of Copernicus, this teaching was persistently called Pythagorean.(Encyclopedic Dictionary of a Young Mathematician: E-68. A. P. Savin.- M.: Pedagogy, 1989, p. 28.)

Some fundamental concepts undoubtedly belongto Pythagoras himself. The first one- the idea of ​​space as mathematicsa tically ordered whole. Pythagoras came to him after discovering that the fundamental harmonic intervals, i.e., octave, perfect fifth and perfect fourth, arise when the lengths of vibrating strings are related as 2:1, 3:2 and 4:3 (legend has it that the discovery was made whenPythagoras passed by a forge: anvils with different massesgenerated the corresponding sound relationships upon impact). UsmotRevealing an analogy between the orderliness in music, expressed by the relationships discovered by it, and the orderliness of the material world, Pythagorascame to the conclusion that it is permeated with mathematical relationshipsthe whole space. An attempt to apply the mathematical discoveries of Pythagoras to speculative physical constructions led to interesting consequences.results. Thus, it was assumed that each planet during its revolutionaround the Earth it emits as it passes through the clear upper air, or "ether",tone of a certain pitch. The pitch of the sound changes depending on the speedspeed of the planet's movement, the speed depends on the distance to the Earth. PlumWhen celestial sounds come together, they form what is called the “harmony of the spheres,” or “music of the spheres,” references to which are frequent in European literature.

The early Pythagoreans believed that the Earth was flat and in the centerspace. Later they began to believe that the Earth has a spherical shape and, together with other planets (which they included the Sun), is shapedrevolves around the center of space, i.e., the “hearth”.

In antiquity, Pythagoras was best known as a preachersecluded lifestyle. Central to his teaching was the ideatalk about reincarnation (transmigration of souls), which, of course, presupposes the ability of the soul to survive the death of the body, and therefore its immortality. Since in a new incarnation the soul can move into the body of an animal, Pythagoras was opposed to killing animals, eating their meat, and even stated that one should not deal with those who slaughter animals or butcher their carcasses. As far as one can judge from the writings of Empedocles, who shared the religious views of Pythagoras, the shedding of blood was considered here as an original sin, for which the soul is expelled into the mortal world, where it wanders, being imprisoned in one body or another. The soul passionately desires liberation, but out of ignorance it invariably repeats the sinful act.

Can save the soul from an endless series of reincarnationscleansing The simplest cleansing consists in observing certainprohibitions (for example, abstaining from intoxication or drinkingeating beans) and rules of behavior (for example, honoring elders, obeying the law and not being angry).

The Pythagoreans highly valued friendship, and according to their concepts, all the property of friends should be common. A select few were offered the highest form of purification - philosophy, that is, love of wisdom, and therefore the desire for it (this word, according to Cicero, was first used by Pythagoras, who called himself not a sage, but a lover of wisdom). By means of these means the soul comes into contact with the principles of cosmic order and becomes in tune with them, it is freed from its attachment to the body, its lawless and disordered desires. Mathematics is one of the components religionPythagoreans, who taught that God laid the number at the basis of the worldorder.

Influence of the Pythagorean Brotherhood in the first halfVV. BC e. Notincreased continuously. But his desire to give power to the “best” came into conflict with the rise of democratic sentiment in the Greek cities of southern Italy, and soon after 450 BC. e. there was an outbreak in Crotonea rebellion against the Pythagoreans that resulted in the murder and expulsion of many, if not all, members of the brotherhood. However, still inIVV. BC e. pythagoThe Reichs enjoyed influence in southern Italy, and in Tarentum, where Plato’s friend Archytas lived, it remained even longer. However, much more important for the history of philosophy was the creation of Pythagorean centers in Greece itself,for example in Thebes, in the second halfVV. BC e. Hence the Pythagoreanideas penetrated to Athens, where, according to Plato's dialoguePhaedo,they were adopted by Socrates and turned into a broad ideological movement,started by Plato and his student Aristotle.

In subsequent centuries, the figure of Pythagoras himself was surrounded
many legends: he was considered the reincarnated god Apollo,
it was believed that he had a golden thigh and was capable of teaching in
the same time in two places. Early Christian Church Fathers answer
whether Pythagoras has a place of honor between Moses and Plato. Back inXVIV[
there were frequent references to the authority of Pythagoras in matters not only of science |.:
but also magic.(Electronic encyclopedia:StarWorld.).

Behind the legend is the truth

The discovery of the Pythagorean theorem is surrounded by a halo of beautiful legendsProclus, commenting on the last sentenceIbooks "Elements" by Euclid,writes: “If you listen to those who like to repeat ancient legends, thenwe have to say that this theorem goes back to Pythagoras; they saythat he sacrificed a bull in honor of this.” This legend has firmly grown togetherwith the Pythagorean theorem and after 2000 years continued to cause hot clicks. Thus, the optimist Mikhailo Lomonosov wrote: “Pythagoras for the invention of one geometricAccording to the rule of Zeus, he sacrificed a hundred oxen.But if for those found in modern times fromwitty mathematicians rules according to his superstitiousjealousy to act, then barelyif there were so many in the whole worldcattle have been found."

But the ironic Heinrich Heine saw the development of the same situation somewhat differently : « Who knows ! Who knows ! Maybe , the soul of Pythus the mountain moved into the poor candidate , who could not prove the Pythagorean theorem and failed from - for this in exams , while in his examiners dwell the souls of those bulls , which Pythagoras , delighted by the discovery of his theorem , sacrificed to the immortal gods ».

History of the discovery of the theorem

The discovery of the Pythagorean theorem is usually attributed to the ancient Greek philosopher and mathematician Pythagoras (VIV. BC e.). But a study of Babylonian cuneiform tables and ancient Chinese manuscripts (copies of even more ancient manuscripts) showed that this statement was known long before Pythagoras, perhaps millennia before him. The merit of Pythagoras was that he discovered the proof of this theorem.

Historical overview Let's start with ancient China. There is a special note heremania is attracted by the mathematical book Chu-pei. This work talks about the Pythagorean triangle with sides 3, 4 and 5:“If a right angle is decomposed into its component parts, then the line connecting the ends of its sides will be 5, when the base is 3 and the height is 4.”

In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.

Also, the Pythagorean theorem was discovered in the ancient Chinese treatise “Zhou-bi suan jin” (“Mathematical treatiseabout the gnomon"), the time of creation of which is unknown exactly, but where it is stated that inXVV. BC e. the Chinese knew the properties of the Egyptian triangle, and inXVIV. BC e. - and the general form of the theorem.

Cantor (the greatest German historian of mathematics) believes that the equality 3 2 + 4 2 = 5 2 was already known to the Egyptians around 2300 BC. e. during the time of King AmenemhetI(according to papyrus 6619 of the Berlin Museum).

According to Cantor, harpedonaptes, or “rope pullers,” built right angles when

using right triangles with sides 3, 4 and 5.

It is very easy to reproduce their methodconstruction. Let's take a rope 12 m long and tie a colored stripe to it at a distance3 m from one end and 4 m from the other. Right anglewill be enclosed between sides 3 and 4 m long. It could be objected to the Harpedonaptes that their method of construction becomes redundant if one uses, for example, a wooden square, which is used by all carpenters. Indeed, Egyptian drawings are known in which such a tool is found, for example, drawings depicting a carpenter's workshop.Somewhat more is known aboutPythagorean theorem among the Babylonians.In one text dating back to the timeMeni Hammurabi, i.e. by 2000BC e., an approximate calculation of the hypotenuse is given directlycoal triangle. From herewe can conclude that in Dvurawho knew how to do calculationswith right trianglesmi, at least in somecases. Based on onesides, at today's levelknowledge about Egyptian and Babylonianmathematics, and on the other - in criticismlogical study of Greek sources, Van der Waerden (DutchRussian mathematician) made the following conclusion:

“The merit of the first Greek mathematicians, such as Thales, Pythagoras and the Pythagoreans, is not the discovery of mathematics, but its systematization and justification. The computational recipe is in their hands you, based on vague ideas, have turned into precise new science."

The geometry of the Hindus, like that of the Egyptians and Babylonians, was closelyassociated with a cult. It is very likely that the square theorem is hypotenuse was known in India for aboutXVIIIcentury BC e., alsoit was also known in ancient Indian geometrictheological treatiseVII- Vcenturies BC e. "Sulva Sutra" ("Rulesropes").

But despite all this evidence, the name of Pythagoras is sofirmly fused with the Pythagorean theorem, which is simply impossible nowone can imagine that this phrase will fall apart. Same fromalso refers to the legend of the spell of Pythagoras' bulls. And it's unlikelyneed to be dissected with a historical-mathematical scalpeldeep ancient legends.

Methods to prove the theorem

Proof of the Pythagorean Theorem by Middle Ages Studentsconsidered very difficult and called itDons asinorum - donkey bridge, orelefuga - flight of the “poor”, as some “poor” students who did not have serious mathematical training fledwhether from geometry. Weak students who have memorized theoremswithout understanding and therefore nicknamed “donkeys”, were unableability to overcome the Pythagorean theorem, which seemed to serve themsurmountable bridge. Because of the drawings accompanying the theoremPythagoras, students also called her “ windmill", withthey wrote poems like “Pythagorean trousers are equal on all sides” and drew cartoons.

A). The simplest proof

Probably the fact stated in the Pythagorean theorem was a dreamchala is set for isosceles rectangles. Just look at the mosaic of black and light triangles,to verify the validity of the theorem for triangleska ABC : a square built on the hypotenuse contains four triangles, and on each side a square is built containingtwo triangles (Fig. 1, 2).

Proofs based on the use of the concept of equal size of figures.

In this case, we can consider evidence in which quadrath built on the hypotenuse of a given rectangular trianglesquare, “made up” of the same figures as the squares built on the legs. The following evidence can also be consideredva, in which the permutation of summand figures anda number of new ideas are taken into account.

In Fig. 3 shows two equal squares. Length of sides eachequal to the squarea + b. Each of the squares is divided into parts,consisting of squares and right triangles. It is clear that if you subtract quadruple the area of ​​a right triangle with legs from the area of ​​a squarea, b, then they will remain equal have mercy, i.e. With 2 = a 2 + b 2 . However, the ancient Hindus, who belonged tothis reasoning lies, usually they did not write it down, but accompanied itdrawing with just one word: “Look!” It is quite possible that shePythagoras also offered some proof.

b). Evidence by the method of completion.

The essence of this method is that to the squares, constructon the legs, and to a square built on the hypotenuse, withconnect equal figures so that they are equalnew figures.

In Fig. 4 shows a regular Pythagorow figure right triangleABCwith squares built on its sides. Attached to this figure are threesquares 1 and 2, equal to the original straightcoal triangle.

The validity of the Pythagorean theorem follows from the equal size of hexagonsAEDFPB And ACBNMQ. Here is a direct EP delit hexagonAEDFPBinto two equal quadrilaterals, line CM divides the hexagonACBNMQinto two equal quadrangles; rotating the plane 90° around center A maps the quadrilateral AERB onto a quadrilateralACMQ.

(This proof was first given by Leonard before da Vinci.)

Pythagorean figure completedto a rectangle whose sides are parallelaligned with the corresponding sides of the quadracoms built on legs. Let's divide this rectangle into triangles and straightsquares. From the resulting rectangleFirst, we subtract all the polygons 1, 2, 3, 4, 5, 6, 7, 8, 9, leaving a square built on the hypotenuse. Then from the same rectangle we subtract rectangles 5, 6, 7 and shaded straightsquares, we get squares built on the legs.

Now let us prove that the figures subtracted in the first case areare equal in size to the figures subtracted in the second case.

This illustrates the proof,given by Nassir-ed-Din (1594). Here: P.L.- straight;

KLOA = ACPF = ACED = a 2 ;

LGBO= SVMR = CBNQ = b 2 ;

AKGB = AKLO + LGBO= c 2 ;

hence with 2 = a 2 + b 2 .

Rice. 7 illustrates the proof,given by Hoffmann (1821). HereThe Pythagorean figure is constructed in such a way thatsquares lie on one side of a lineAB. Here:

OCLP = ACLF = ACED = b 2 ;

CBML=CBNQ= A 2 ;

OVMR =ABMF= With 2 ;

OVMR = OCLP + CBML;

Hence c 2 = a 2 + b.

This illustrates another more original evidence offeredHoffman. Here: triangleABC with straight wash angle C; segmentB.F.perpendicularNE and equal to it, segmentBEperpendicularAB and equal to it, segmentAD perpendicular ren AC and equal to it; pointsF, WITH, D belongs reap one straight line; quadrilateralsADFBand ACVE are equal in size, sinceABF= ESV; trianglesADF And ACEs are equal in size;

subtract from both equal quadranglesnicks have a common triangleABC, we get ½ a* a + ½ b* b – ½ c* c

V). Algebraic method of proof.

The figure illustrates the proof of the great Indian mathematician Bhaskari (the famous author of Li-lavati,XIIV.). The drawing was accompanied by only one word: LOOK! Among the proofs of the Pythagorean theorem by the algebraic method, first place (perhaps the oldest) fortakes evidence using subtext bee.

Historians believe that Bhaskara was born sting area with 2 square built onhypotenuse, as the sum of the areas of four triangles 4(ab/2) and the area of ​​a square with a side equal to the difference of the legs.

Let us present in a modern presentation one of these proofs:bodies belonging to Pythagoras.

I "

In Fig. 10 ABC - rectangular, C - right angle, ( C.M.L AB) b - leg projection b to the hypotenuse, A - leg projectionA on the hypotenuse, h - altitude of the triangle drawn to hypotenuse. From the fact that ABC is similar to AFM, it followsb 2 = cb; (1) from the fact that ABC is similar to VSM, it follows that 2 = CA (2) Adding equalities (1) and (2) term by term, we obtain a 2 + b 2 = cb + ca = = c (b + a) = c 2 .

If Pythagoras actually offered such a proof,then he was familiar with a number of important geometric theorems,which modern historians of mathematics usually attribute Euclid.

Proof of Möhl- manna. Area given right trianglenika, on the one hand, is equal to 0,5 a* b, on the other 0.5* p*g, where p - semiperimeter of a triangler - radius inscribed in it is approx.circumference (r = 0.5 - (a + b - c)).We have: 0.5*a*b - 0.5*p*g - 0.5 (a + b + c) * 0.5-(a + b - c), from where it follows that c 2 = a 2 + b 2 .

d) Garfield's proof.

In Figure 12 there are three straighttriangles form a trapezoid. That's why.the area of ​​this figure is possible.\ find using the area formuladi rectangular trapezoid,or as the sum of areasthree triangles. In the laneIn this case, this area is equal toby 0.5 (a + b) (a + b), in second rum - 0.5* a* b+ 0.5*a* b+ 0.5*s 2

Equating these expressions, we obtain the Pythagorean theorem.

There are many proofs of the Pythagorean theorem, carried outusing both each of the described methods and using a combinationnia various methods. Concluding the review of examples of various docksstatements, here are some more drawings illustrating the eight waysbov, to which there are references in Euclid’s “Elements” (Fig. 13 - 20).In these drawings the Pythagorean figure is depicted as a solid lineher, and additional constructions - dotted.

As mentioned above, the ancient Egyptians for more than 2000 yearsago, they practically used the properties of a triangle with sides 3, 4, 5 to construct a right angle, that is, they actually used the theorem inverse to the Pythagorean theorem. Let us present a proof of this theorem based on the criterion for the equality of triangles (i.e., one that can be introduced very early into schoolnew practice). So let the sides of the triangleABC (Fig. 21) related to 2 = a 2 + b 2 . (3)

Let us prove that this triangle is right-angled.

Let's construct a right triangleA B C on two sides, whose lengths are equal to the lengthsA And b legs of a given triangle. Let the length of the hypotenuse of the constructed triangle be on c . Since the constructed triangle is right-angled, then by theoryin the Pythagorean rheme we havec = a + b (4)

Comparing relations (3) and (4), we obtain thatWith= with or c = c Thus, the triangles - the given one and the one constructed - are equal, since they have three respectively equal sides. Angle Cis straight, therefore angle C of this triangle is also right.

Additive evidence.

These proofs are based on the decomposition of squares built on the sides into figures from which a quad can be formedrath built on the hypotenuse.

Einstein's proof ( rice. 23) based on decompositiona square built on the hypotenuse into 8 triangles.

Here: ABC- rectangular triangle with right angle C;COMN; SK MN; P.O.|| MN; E.F.|| MN.

Prove it yourselftrue equality of triangles, halfcalculated by dividing the squares according tobuilt on legs and hypotenuse.

b) Based on the proof of al-Nayriziyah, another decomposition of squares into pairwise equal figures was carried out (hereABC - right triangle with right angle C).

This proof is also called “hinged” becausethat here only two parts, equal to the original triangle, change their position, and they are, as it were, attached to the restfigure on hinges around which they rotate (Fig. 25).

c) Another proof by the method of decomposing squares intoequal parts, called a "wheel with blades", is shown in rice. 26. Here: ABC - right triangle with right angle scrap S, O - the center of a square built on a large side; dotted lines passing through a pointABOUT, perpendicular orparallel to the hypotenuse.

This decomposition of squares is interesting because its pairwise equal quadrilaterals can be mapped onto each other by parallel translation.

"Pythagorean trousers" (Euclid's proof).

For two thousand yearschanged the proof inventedEuclid, which is placed in histhe famous "Principles". Euclid opus cal height VN from the vertex of a right triangle to the hypotenuse and proved that its continuation divides the square constructed on the hypotenuse into two rectangles whose areas are equal

areas of the corresponding squares built on the sides. Euclid's proof in comparison with the ancient Chinese or ancient Indian looks likeoverly complicated. For this reasonhe was often called “stilted” and “contrived.” But this opinionsuperficial. The drawing used to prove the theorem is jokingly called “Pythagorean pants.” Forfor a long time it was considered one of the symbols of mathematical science.

Ancient Chinese evidence.

Mathematical treatises Ancient China reached us in the editorial officeIIV. BC e. The fact is that in 213 BC. e. chinese emperor

Shi Huangdi, trying to eliminate previous traditions, ordered all ancient books to be burned. InIIV. BC e. Paper was invented in China and at the same time the restoration beganancient books. This is how “Mathematics in Nine Books” arose -the main surviving mathematical and astronomical works ny.

In the 9th book of "Mathematics" there is a drawingwho proves the Pythagorean theorem.The key to this proof is not difficult to find (Fig. 27).

In fact, in ancient Chinesethe same four equal rectangular trianglessquare with legsa, c and hypotenuse With laid so that their outer contour isthere is a square with a sidea + b, and internal - a square with side c, built on the hypotenuse (Fig. 28).

If a square with sideWith cut and the remaining 4 shaded trianglesplaced in two rectangles, it is clear that the resulting void, on the one hand,

equal to With, and on the other

a + b 2 , i.e. With 2 = a 2 + b

The theorem has been proven.

Note that with such a proof

Constructions inside the square on the hypotenwe see
dim in the ancient Chinese drawing are not used (Fig. 30). Apparently, ancient Chinese mathematicians had something different beforeproof, namely: if squared with
sideWith two shaded trianglescut off the nick and attach the hypotenuses totwo other hypotenuses, then it is easy to findconfirm that the resulting figure, which sometimes called the "bride's chair", withconsists of two squares with sidesA Andb, i.e. with 2 = A 2 + b 2 .

The figure reproduces blackfrom the treatise “Zhou-bi...”. HerePythagorean theorem considered forEgyptian triangle with legs3, 4 and hypotenuse 5 units of measurement.The square on the hypotenuse contains 25cells, and the square inscribed in it on the larger side is 16. It is clear that the remaining part contains 9 cells. This andthere will be a square on the smaller side.

Around and around

The history of the Pythagorean theorem goes back centuries and millennia. In this article, we will not dwell in detail on historical topics. For the sake of intrigue, let’s just say that, apparently, this theorem was known to the ancient Egyptian priests who lived more than 2000 years BC. For those curious, here is a link to the Wikipedia article.

First of all, for the sake of completeness, I would like to present here the proof of the Pythagorean theorem, which, in my opinion, is the most elegant and obvious. The picture above shows two identical squares: left and right. It can be seen from the figure that on the left and right the areas of the shaded figures are equal, since in each of the large squares there are 4 identical right triangles shaded. This means that the unshaded (white) areas on the left and right are also equal. We note that in the first case the area of ​​the unshaded figure is equal to , and in the second case the area of ​​the unshaded region is equal to . Thus, . The theorem is proven!

How to call these numbers? You can’t call them triangles, because four numbers can’t form a triangle. And here! Like a bolt from the blue

Since there are such quadruples of numbers, it means there must be a geometric object with the same properties reflected in these numbers!

Now all that remains is to select some geometric object for this property, and everything will fall into place! Of course, the assumption was purely hypothetical and had no basis in support. But what if this is so!

The selection of objects has begun. Stars, polygons, regular, irregular, right angle, and so on and so forth. Again nothing fits. What to do? And at this moment Sherlock gets his second lead.

We need to increase the size! Since three corresponds to a triangle on a plane, then four corresponds to something three-dimensional!

Oh no! Too many options again! And in three dimensions there are much, much more different geometric bodies. Try to go through them all! But it's not all bad. There is also a right angle and other clues! What do we have? Egyptian fours of numbers (let them be Egyptian, they need to be called something), a right angle (or angles) and some three-dimensional object. Deduction worked! And... I believe that quick-witted readers have already realized that we are talking about pyramids in which, at one of the vertices, all three angles are right. You can even call them rectangular pyramids similar to a right triangle.

New theorem

So, we have everything we need. Rectangular (!) pyramids, side facets and secant face-hypotenuse. It's time to draw another picture.

The picture shows a pyramid with its vertex at the origin of rectangular coordinates (the pyramid seems to be lying on its side). The pyramid is formed by three mutually perpendicular vectors plotted from the origin along the coordinate axes. That is, each side edge A pyramid is a right triangle with a right angle at the origin. The ends of the vectors define the cutting plane and form the base face of the pyramid.

Theorem

Let there be a rectangular pyramid formed by three mutually perpendicular vectors, the areas of which are equal to - , and the area of ​​the hypotenuse face is - . Then

Alternative formulation: For a tetrahedral pyramid, in which at one of the vertices all plane angles are right, the sum of the squares of the areas of the lateral faces is equal to the square of the area of ​​the base.

Of course, if the usual Pythagorean theorem is formulated for the lengths of the sides of triangles, then our theorem is formulated for the areas of the sides of the pyramid. Proving this theorem in three dimensions is very easy if you know a little vector algebra.

Proof

Let's express the areas in terms of the lengths of the vectors.

Where .

Let's imagine the area as half the area of ​​a parallelogram built on the vectors and

As is known, vector product two vectors is a vector whose length is numerically equal to the area of ​​the parallelogram constructed on these vectors.
That's why

Thus,

Q.E.D!

Of course, as a person professionally engaged in research, this has already happened in my life, more than once. But this moment was the brightest and most memorable. I experienced the full range of feelings, emotions, and experiences of a discoverer. From the birth of a thought, the crystallization of an idea, the discovery of evidence - to the complete misunderstanding and even rejection that my ideas met with among my friends, acquaintances and, as it seemed to me then, the whole world. It was unique! I felt like I was in the shoes of Galileo, Copernicus, Newton, Schrödinger, Bohr, Einstein and many many other discoverers.

Afterword

In life, everything turned out to be much simpler and more prosaic. I was late... But by how much! Just 18 years old! Under terrible prolonged torture and not the first time, Google admitted to me that this theorem was published in 1996!

The article was published by the Texas Press technical university. The authors, professional mathematicians, introduced terminology (which, by the way, largely coincided with mine) and also proved a generalized theorem that is valid for a space of any dimension greater than one. What happens in dimensions higher than 3? Everything is very simple: instead of faces and areas there will be hypersurfaces and multidimensional volumes. And the statement, of course, will remain the same: the sum of the squares of the volumes of the side faces is equal to the square of the volume of the base - just the number of faces will be greater, and the volume of each of them will be equal to half the product of the generating vectors. It's almost impossible to imagine! One can only, as philosophers say, think!

Surprisingly, when I learned that such a theorem was already known, I was not at all upset. Somewhere in the depths of my soul, I suspected that it was quite possible that I was not the first, and I understood that I needed to always be prepared for this. But the emotional experience that I received lit a spark of researcher in me, which, I am sure, will now never fade!

P.S.

An erudite reader sent a link in the comments
De Gois' theorem

Excerpt from Wikipedia

In 1783, the theorem was presented to the Paris Academy of Sciences by the French mathematician J.-P. de Gois, but it was previously known to René Descartes and before him Johann Fulgaber, who was probably the first to discover it in 1622. In more general view the theorem was formulated by Charles Tinsault (French) in a report to the Paris Academy of Sciences in 1774

So I was not 18 years late, but at least a couple of centuries late!

Sources

Readers provided several useful links in the comments. Here are these and some other links:

According to Van der Waerden, it is very likely that the ratio in general form was known in Babylon around the 18th century BC. e.

Around 400 BC. BC, according to Proclus, Plato gave a method for finding Pythagorean triplets, combining algebra and geometry. Around 300 BC. e. The oldest axiomatic proof of the Pythagorean theorem appeared in Euclid's Elements.

Formulations

The basic formulation contains algebraic operations - in a right triangle, the lengths of which are equal a (\displaystyle a) And b (\displaystyle b), and the length of the hypotenuse is c (\displaystyle c), the following relation is satisfied:

.

An equivalent geometric formulation is also possible, resorting to the concept of area of ​​a figure: in a right triangle, the area of ​​the square built on the hypotenuse is equal to the sum of the areas of the squares built on the legs. The theorem is formulated in this form in Euclid’s Elements.

Converse Pythagorean theorem- a statement about the rectangularity of any triangle, the lengths of the sides of which are related by the relation a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)). As a consequence, for any triple positive numbers a (\displaystyle a), b (\displaystyle b) And c (\displaystyle c), such that a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)), there is a right triangle with legs a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c).

Proof

There are at least 400 proofs of the Pythagorean theorem recorded in the scientific literature, which is explained by both its fundamental significance for geometry and the elementary nature of the result. The main directions of proofs are: algebraic use of relations between the elements of a triangle (for example, the popular method of similarity), the method of areas, there are also various exotic proofs (for example, using differential equations).

Through similar triangles

The classical proof of Euclid is aimed at establishing the equality of areas between rectangles formed by dissecting the square above the hypotenuse by the height of the right angle with the squares above the legs.

The construction used for the proof is as follows: for a right triangle with a right angle C (\displaystyle C), squares over the legs and and squares over the hypotenuse A B I K (\displaystyle ABIK) height is being built CH and the ray that continues it s (\displaystyle s), dividing the square above the hypotenuse into two rectangles and . The proof aims to establish the equality of the areas of the rectangle A H J K (\displaystyle AHJK) with a square over the leg A C (\displaystyle AC); the equality of the areas of the second rectangle, constituting the square above the hypotenuse, and the rectangle above the other leg is established in a similar way.

Equality of areas of a rectangle A H J K (\displaystyle AHJK) And A C E D (\displaystyle ACED) is established through the congruence of triangles △ A C K ​​(\displaystyle \triangle ACK) And △ A B D (\displaystyle \triangle ABD), the area of ​​each of which is equal to half the area of ​​the squares A H J K (\displaystyle AHJK) And A C E D (\displaystyle ACED) accordingly, in connection with the following property: the area of ​​a triangle is equal to half the area of ​​a rectangle if the figures have a common side, and the height of the triangle to the common side is the other side of the rectangle. The congruence of triangles follows from the equality of two sides (sides of squares) and the angle between them (composed of a right angle and an angle at A (\displaystyle A).

Thus, the proof establishes that the area of ​​a square above the hypotenuse, composed of rectangles A H J K (\displaystyle AHJK) And B H J I (\displaystyle BHJI), is equal to the sum of the areas of the squares over the legs.

Proof of Leonardo da Vinci

The area method also includes a proof found by Leonardo da Vinci. Let a right triangle be given △ A B C (\displaystyle \triangle ABC) with right angle C (\displaystyle C) and squares A C E D (\displaystyle ACED), B C F G (\displaystyle BCFG) And A B H J (\displaystyle ABHJ)(see picture). In this proof on the side HJ (\displaystyle HJ) last time outside a triangle is constructed, congruent △ A B C (\displaystyle \triangle ABC), moreover, reflected both relative to the hypotenuse and relative to the height to it (that is, J I = B C (\displaystyle JI=BC) And H I = A C (\displaystyle HI=AC)). Straight C I (\displaystyle CI) splits the square built on the hypotenuse into two equal parts, since triangles △ A B C (\displaystyle \triangle ABC) And △ J H I (\displaystyle \triangle JHI) equal in construction. The proof establishes the congruence of quadrilaterals C A J I (\displaystyle CAJI) And D A B G (\displaystyle DABG), the area of ​​each of which turns out to be, on the one hand, equal to the sum of half the areas of the squares on the legs and the area of ​​the original triangle, on the other hand, half the area of ​​the square on the hypotenuse plus the area of ​​the original triangle. In total, half the sum of the areas of the squares over the legs is equal to half the area of ​​the square over the hypotenuse, which is equivalent to the geometric formulation of the Pythagorean theorem.

Proof by the infinitesimal method

There are several proofs using the technique of differential equations. In particular, Hardy is credited with a proof using infinitesimal increments of legs a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c), and preserving similarity with the original rectangle, that is, ensuring the fulfillment of the following differential relations:

d a d c = c a (\displaystyle (\frac (da)(dc))=(\frac (c)(a))), d b d c = c b (\displaystyle (\frac (db)(dc))=(\frac (c)(b))).

Using the method of separating variables, a differential equation is derived from them c d c = a d a + b d b (\displaystyle c\ dc=a\,da+b\,db), whose integration gives the relation c 2 = a 2 + b 2 + C o n s t (\displaystyle c^(2)=a^(2)+b^(2)+\mathrm (Const) ). Application of initial conditions a = b = c = 0 (\displaystyle a=b=c=0) defines the constant as 0, which results in the statement of the theorem.

The quadratic dependence in the final formula appears due to the linear proportionality between the sides of the triangle and the increments, while the sum is associated with independent contributions from the increment of different legs.

Variations and generalizations

Similar geometric shapes on three sides

An important geometric generalization of the Pythagorean theorem was given by Euclid in the Elements, moving from the areas of squares on the sides to the areas of arbitrary similar geometric shapes: the sum of the areas of such figures built on the legs will be equal to the area of ​​a similar figure built on the hypotenuse.

The main idea of ​​this generalization is that the area of ​​such a geometric figure is proportional to the square of any of its linear size and in particular the square of the length of any side. Therefore, for similar figures with areas A (\displaystyle A), B (\displaystyle B) And C (\displaystyle C), built on legs with lengths a (\displaystyle a) And b (\displaystyle b) and hypotenuse c (\displaystyle c) Accordingly, the following relation holds:

A a 2 = B b 2 = C c 2 ⇒ A + B = a 2 c 2 C + b 2 c 2 C (\displaystyle (\frac (A)(a^(2)))=(\frac (B )(b^(2)))=(\frac (C)(c^(2)))\,\Rightarrow \,A+B=(\frac (a^(2))(c^(2) ))C+(\frac (b^(2))(c^(2)))C).

Since according to the Pythagorean theorem a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)), then done.

In addition, if it is possible to prove, without invoking the Pythagorean theorem, that for the areas of three similar geometric figures on the sides of a right triangle, the following relation holds: A + B = C (\displaystyle A+B=C), then using the reverse of the proof of Euclid's generalization, one can derive a proof of the Pythagorean theorem. For example, if on the hypotenuse we construct a right triangle congruent with the initial one with an area C (\displaystyle C), and on the sides - two similar right-angled triangles with areas A (\displaystyle A) And B (\displaystyle B), then it turns out that triangles on the sides are formed as a result of dividing the initial triangle by its height, that is, the sum of the two smaller areas of the triangles is equal to the area of ​​the third, thus A + B = C (\displaystyle A+B=C) and, applying the relation for similar figures, the Pythagorean theorem is derived.

Cosine theorem

Pythagoras' theorem is a special case of the more general cosine theorem, which relates the lengths of the sides in an arbitrary triangle:

a 2 + b 2 − 2 a b cos ⁡ θ = c 2 (\displaystyle a^(2)+b^(2)-2ab\cos (\theta )=c^(2)),

where is the angle between the sides a (\displaystyle a) And b (\displaystyle b). If the angle is 90°, then cos ⁡ θ = 0 (\displaystyle \cos \theta =0), and the formula simplifies to the usual Pythagorean theorem.

Free Triangle

There is a generalization of the Pythagorean theorem to an arbitrary triangle, operating solely on the ratio of the lengths of the sides, it is believed that it was first established by the Sabian astronomer Thabit ibn Qurra. In it, for an arbitrary triangle with sides, an isosceles triangle with a base on the side fits into it c (\displaystyle c), the vertex coinciding with the vertex of the original triangle, opposite the side c (\displaystyle c) and angles at the base equal to the angle θ (\displaystyle \theta ), opposite side c (\displaystyle c). As a result, two triangles are formed, similar to the original one: the first - with sides a (\displaystyle a), the side farthest from it of the inscribed isosceles triangle, and r (\displaystyle r)- side parts c (\displaystyle c); the second - symmetrically to it from the side b (\displaystyle b) with the side s (\displaystyle s)- the corresponding part of the side c (\displaystyle c). As a result, the following relation is satisfied:

a 2 + b 2 = c (r + s) (\displaystyle a^(2)+b^(2)=c(r+s)),

degenerating into the Pythagorean theorem at θ = π / 2 (\displaystyle \theta =\pi /2). The relationship is a consequence of the similarity of the formed triangles:

c a = a r , c b = b s ⇒ c r + c s = a 2 + b 2 (\displaystyle (\frac (c)(a))=(\frac (a)(r)),\,(\frac (c) (b))=(\frac (b)(s))\,\Rightarrow \,cr+cs=a^(2)+b^(2)).

Pappus's theorem on areas

Non-Euclidean geometry

The Pythagorean theorem is derived from the axioms of Euclidean geometry and is not valid for non-Euclidean geometry - the fulfillment of the Pythagorean theorem is equivalent to the Euclidian parallelism postulate.

In non-Euclidean geometry, the relationship between the sides of a right triangle will necessarily be in a form different from the Pythagorean theorem. For example, in spherical geometry, all three sides of a right triangle, which bound the octant of the unit sphere, have a length π / 2 (\displaystyle \pi /2), which contradicts the Pythagorean theorem.

Moreover, the Pythagorean theorem is valid in hyperbolic and elliptic geometry if the requirement that the triangle is rectangular is replaced by the condition that the sum of two angles of the triangle must be equal to the third.

Spherical geometry

For any right triangle on a sphere with radius R (\displaystyle R)(for example, if the angle in a triangle is right) with sides a , b , c (\displaystyle a,b,c) the relationship between the sides is:

cos ⁡ (c R) = cos ⁡ (a R) ⋅ cos ⁡ (b R) (\displaystyle \cos \left((\frac (c)(R))\right)=\cos \left((\frac (a)(R))\right)\cdot \cos \left((\frac (b)(R))\right)).

This equality can be derived as a special case of the spherical cosine theorem, which is valid for all spherical triangles:

cos ⁡ (c R) = cos ⁡ (a R) ⋅ cos ⁡ (b R) + sin ⁡ (a R) ⋅ sin ⁡ (b R) ⋅ cos ⁡ γ (\displaystyle \cos \left((\frac ( c)(R))\right)=\cos \left((\frac (a)(R))\right)\cdot \cos \left((\frac (b)(R))\right)+\ sin \left((\frac (a)(R))\right)\cdot \sin \left((\frac (b)(R))\right)\cdot \cos \gamma ). ch ⁡ c = ch ⁡ a ⋅ ch ⁡ b (\displaystyle \operatorname (ch) c=\operatorname (ch) a\cdot \operatorname (ch) b),

Where ch (\displaystyle \operatorname (ch) )- hyperbolic cosine. This formula is a special case of the hyperbolic cosine theorem, which is valid for all triangles:

ch ⁡ c = ch ⁡ a ⋅ ch ⁡ b − sh ⁡ a ⋅ sh ⁡ b ⋅ cos ⁡ γ (\displaystyle \operatorname (ch) c=\operatorname (ch) a\cdot \operatorname (ch) b-\operatorname (sh) a\cdot \operatorname (sh) b\cdot \cos \gamma ),

Where γ (\displaystyle \gamma )- an angle whose vertex is opposite to the side c (\displaystyle c).

Using the Taylor series for the hyperbolic cosine ( ch ⁡ x ≈ 1 + x 2 / 2 (\displaystyle \operatorname (ch) x\approx 1+x^(2)/2)) it can be shown that if a hyperbolic triangle decreases (that is, when a (\displaystyle a), b (\displaystyle b) And c (\displaystyle c) tend to zero), then the hyperbolic relations in a right triangle approach the relation of the classical Pythagorean theorem.

Application

Distance in two-dimensional rectangular systems

The most important application of the Pythagorean theorem is determining the distance between two points in a rectangular coordinate system: distance s (\displaystyle s) between points with coordinates (a , b) (\displaystyle (a,b)) And (c , d) (\displaystyle (c,d)) equals:

s = (a − c) 2 + (b − d) 2 (\displaystyle s=(\sqrt ((a-c)^(2)+(b-d)^(2)))).

For complex numbers, the Pythagorean theorem gives a natural formula for finding the modulus of a complex number - for z = x + y i (\displaystyle z=x+yi) it is equal to the length

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Introduction

1. From the biography of Pythagoras

2. Pythagoras and the Pythagoreans

3. From the history of the creation of the theorem

4. Six proofs of the theorem

4.1 Ancient Chinese evidence

4.2 Proof by J. Hardfield

4.3 Proof is the oldest

4.4 The simplest proof

4.5 Proof of the ancients

4.6 Euclid's proof

5. Application of the Pythagorean theorem

5.1 Theoretical problems

5.2 Practical problems (old)

Conclusion

References

INTRODUCTION

In this academic year we became acquainted with an interesting theorem, known, as it turned out, since ancient times:

« Square,builtonhypotenuserectangulartriangleequal sizeamountsquaresbuiltonlegs»

The discovery of this statement is usually attributed to the ancient Greek philosopher and mathematician Pythagoras (6th century BC). But the study of ancient manuscripts showed that this statement was known long before the birth of Pythagoras.

We wondered why, in this case, it is associated with the name of Pythagoras.

The purpose of our research was to find out who Pythagoras was and how he relates to this theorem. Studying the history of the theorem, we decided to find out:

Are there other proofs of this theorem?

What is the significance of this theorem in people's lives?

What role did Pythagoras play in the development of mathematics?

1. From the biography of Pythagoras

Pythagoras of Samos is a great Greek scientist. His name is familiar to every schoolchild. If you are asked to name one ancient mathematician, the vast majority will name Pythagoras. His fame is associated with the name of the Pythagorean theorem. Although we now know that this theorem was known in ancient Babylon 1200 years before Pythagoras, and in Egypt 2000 years before him a right triangle with sides 3, 4, 5 was known, we still call it by the name of this ancient scientist.

Almost nothing is known reliably about the life of Pythagoras, but his name is associated large number legends.

Pythagoras was born in 570 BC. e on the island of Samos. Pythagoras's father was Mnesarchus, a carver precious stones. Mnesarchus, according to Apuleius, “was famous among the craftsmen for his art of cutting gems,” but acquired fame rather than wealth. The name of Pythagoras' mother has not been preserved.

Pythagoras had a beautiful appearance, wore a long beard, and a golden diadem on his head. Pythagoras is not a name, but a nickname that the philosopher received because he always spoke correctly and convincingly, like a Greek oracle. (Pythagoras - “persuasive by speech”).

Among the teachers of young Pythagoras were the elder Hermodamantus and Pherecydes of Syros (although there is no firm certainty that it was Hermodamantus and Pherecydes who were Pythagoras’s first teachers). Young Pythagoras spent whole days at the feet of the elder Hermodamas, listening to the melody of the cithara and the hexameters of Homer. Pythagoras retained his passion for the music and poetry of the great Homer throughout his life. And, being a recognized sage, surrounded by a crowd of disciples, Pythagoras began the day by singing one of Homer’s songs.

Pherecydes was a philosopher and was considered the founder of the Italian school of philosophy. Thus, if Hermodamant introduced young Pythagoras into the circle of muses, then Pherecydes turned his mind to logos. Pherecydes directed Pythagoras’s gaze to nature and advised him to see his first and main teacher in nature.

But be that as it may, the restless imagination of young Pythagoras very soon became cramped in little Samos, and he went to Miletus, where he met another scientist - Thales. Thales advised him to go to Egypt for knowledge, which Pythagoras did.

In 550 BC, Pythagoras makes a decision and goes to Egypt. So, an unknown country and an unknown culture open up before Pythagoras. Much amazed and surprised Pythagoras in this country, and after some observations of the life of the Egyptians, Pythagoras realized that the path to knowledge, protected by the priestly caste, lay through religion.

Together with the Egyptian boys, he, a mature Ellin with a black curly beard, sat down at the limestone plates. But unlike his smaller comrades, the bearded Ellin’s ears were not on his back, and his head stood still. Very soon Pythagoras far surpassed his classmates. But the school of scribes was only the first step on the path to secret knowledge.

After eleven years of study in Egypt, Pythagoras goes to his homeland, where along the way he ends up in Babylonian captivity. There he becomes acquainted with Babylonian science, which was more developed than Egyptian. The Babylonians were able to solve linear, quadratic, and some types of cubic equations. They successfully applied the Pythagorean theorem more than 1000 years before Pythagoras. Having escaped from captivity, he was unable to stay in his homeland for long due to the atmosphere of violence and tyranny that reigned there. He decided to move to Croton (a Greek colony in northern Italy).

It was in Croton that the most glorious period in the life of Pythagoras began. There he established something like a religious-ethical brotherhood or a secret monastic order, the members of which were obliged to lead the so-called Pythagorean way of life.

2. PythagorasAndPythagoreans

Pythagoras organized in Greek colony in the south of the Apennine Peninsula, a religious and ethical brotherhood, such as a monastic order, which would later be called the Pythagorean Union. Members of the union had to adhere to certain principles: firstly, to strive for the beautiful and glorious, secondly, to be useful, and thirdly, to strive for high pleasure.

The system of moral and ethical rules, bequeathed by Pythagoras to his students, was compiled into a peculiar moral code of the Pythagoreans “Golden Verses”, which were very popular in the era of Antiquity, the Middle Ages and the Renaissance. The Pythagorean system of classes consisted of three sections:

teaching about numbers - arithmetic,

teachings about figures - geometry,

doctrines about the structure of the Universe - astronomy.

The education system founded by Pythagoras lasted for many centuries.

The Pythagoreans taught that God placed numbers at the basis of the world order. God is unity, and the world is plurality and consists of opposites. That which brings opposites to unity and connects everything into the cosmos is harmony. Harmony is divine and lies in numerical expressions. Whoever studies harmony to the end will himself become divine and immortal.

Music, harmony and numbers were inextricably linked in the teachings of the Pythagoreans. Mathematics and numerical mysticism were fantastically mixed in him. Pythagoras believed that number is the essence of all things and that the Universe is a harmonious system of numbers and their relationships.

The Pythagorean school did a lot to give geometry the character of a science. The main feature of the Pythagorean method was the combination of geometry with arithmetic.

Pythagoras dealt a lot with proportions and progressions and, probably, with the similarity of figures, since he is credited with solving the problem: “Given two figures, construct a third, equal in size to one of the data and similar to the second.”

Pythagoras and his students introduced the concept of polygonal, friendly, perfect numbers and studied their properties. Pythagoras was not interested in arithmetic as a practice of calculation, and he proudly declared that he “put arithmetic above the interests of the merchant.”

Pythagoras was one of the first to believe that the Earth is spherical and is the center of the Universe, that the Sun, Moon and planets have their own movement, different from the daily movement of the fixed stars.

Nicolaus Copernicus perceived the teaching of the Pythagoreans about the movement of the Earth as the prehistory of his heliocentric teaching. No wonder the church declared the Copernican system a “false Pythagorean doctrine.”

In the school of Pythagoras, the discoveries of students were attributed to the teacher, so it is almost impossible to determine what Pythagoras himself did and what his students did.

Disputes have been going on around the Pythagorean Union for the third millennium, but there is still no general consensus. The Pythagoreans had many symbols and signs that were a kind of commandments: for example, “don’t step through the scales,” i.e. do not violate justice; “Don’t stir up fire with a knife,” that is, don’t hurt angry people with offensive words.

But the main Pythagorean symbol - a symbol of health and an identification mark - was the pentagram or Pythagorean star - a star-shaped pentagon formed by the diagonals of a regular pentagon.

Members of the Pythagorean League were residents of many cities in Greece.

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The Pythagoreans also accepted women into their society. The union flourished for more than twenty years, and then persecution of its members began, many of the students were killed.

There has been a lot of speculation about the death of Pythagoras himself. different legends. But the teachings of Pythagoras and his students continued to live.

3. FromhistorytheoremsPythagoras

It is now known that this theorem was not discovered by Pythagoras. However, some believe that it was Pythagoras who first gave its full proof, while others deny him this merit. Some attribute to Pythagoras the proof which Euclid gives in the first book of his Elements. On the other hand, Proclus claims that the proof in the Elements belongs to Euclid himself.

As we see, the history of mathematics has preserved almost no reliable specific data about the life of Pythagoras and his mathematical activities. But the legend even tells us the immediate circumstances that accompanied the discovery of the theorem. Many people know the sonnet of the German novelist Chamisso:

The truth will remain eternal, as soon as

A weak person will know it!

And now the Pythagorean theorem

True, as in his distant age.

The sacrifice was abundant

To the gods from Pythagoras. One Hundred Bulls

He gave it up to be slaughtered and burned

Behind the light is a ray that came from the clouds.

Therefore, ever since then,

The truth is just being born,

The bulls roar, sensing her, following her,

They are unable to stop the light,

Or they can only close their eyes and tremble

From the fear that Pythagoras instilled in them.

We begin our historical review of the Pythagorean theorem with ancientChina. Here special attention I'm attracted to Chu-Pei's math book. This work talks about the Pythagorean triangle with sides 3, 4 and 5:

« Ifdirectcornerspread outoncompositeparts,Thatline,connectingendshissides,will5, WhenbaseThere is3, Aheight4 » .

It is very easy to reproduce their method of construction. Let's take a rope 12 m long and tie a colored strip to it at a distance of 3 m. from one end and 4 meters from the other.

The right angle will be enclosed between sides 3 and 4 meters long. In the same book, a drawing is proposed that coincides with one of the drawings of the Hindu geometry of Bashara.

Cantor(the largest German historian of mathematics) believes that the equality 3І + 4І = 5І was already known to the Egyptians around 2300 BC, during the time of King Amenemhet I (according to papyrus 6619 of the Berlin Museum).

According to Cantor, the harpedonaptes, or “rope pullers,” built right angles using right triangles with sides of 3, 4, and 5.

The Babylonians knew somewhat more about the Pythagorean theorem. In one text dating back to the time of Hammurabi, i.e. by 2000 BC, an approximate calculation of the hypotenuse of a right triangle is given; from this we can conclude that in Mesopotamia they were able to perform calculations with right triangles, at least in some cases.

GeometryatHindus was closely associated with the cult. It is very likely that the square of the hypotenuse theorem was already known in India around the 8th century BC. Along with purely ritual prescriptions, there are also works of a geometric theological nature, called Sulvasutras. In these writings dating back to the 4th or 5th century BC, we encounter the construction of a right angle using a triangle with sides 15, 36, 39.

INaveragecentury Pythagoras' theorem defined the limit, if not the greatest possible, then at least good mathematical knowledge. The characteristic drawing of the Pythagorean theorem, which today is sometimes transformed by schoolchildren, for example, into a professor dressed in a robe or a man in a top hat, was often used in those days as a symbol of mathematics.

In conclusion, we present various formulations of the Pythagorean theorem translated from Greek, Latin and German.

UEuclid This theorem states (literal translation):

INrectangulartrianglesquaresides,tenseoverdirectangle,equalssquaresonsides,concludingdirectcorner.

Latin translation of Arabic text Annaricia(circa 900 BC), made by Gerhard Cremona(12th century) reads (translated):

"Ineveryrectangulartrianglesquare,educatedonside,tenseoverdirectangle,equalsamounttwosquares,educatedontwosides,concludingdirectcorner"

In Geometry Culmonensis (circa 1400) the theorem reads like this (in translation): « So,squaresquare,measuredBylengthside,sosamegreat,Howattwosquares,whichmeasuredBytwopartieshis,adjacentTodirectcorner»

In the Russian translation of Euclidean “Principles”, the Pythagorean theorem is stated as follows: "INrectangulartrianglesquarefromsides,oppositedirectcorner,equalsamountsquaresfromsides,containingdirectcorner".

As we see, in different countries And different languages There are different versions of the formulation of the familiar theorem. Created in different times and in different languages, they reflect the essence of one mathematical law, the proof of which also has several options.

pythagoras mathematics theorem proof

4. SixwaysprooftheoremsPythagoras

4.1 Ancient Chineseproof

An ancient Chinese drawing shows four equal right-angled triangles with legs a, b and hypotenuse With laid so that their outer contour forms a square with side a+ b, and the inner one is a square with side With, built on the hypotenuse

a 2 + 2ab +b 2 = c 2 + 2ab

4.2 ProofJ.Hardfield(1882 G.)

Let's arrange two equal right triangles so that the leg of one of them is a continuation of the other.

The area of ​​the trapezoid under consideration is found as the product of half the sum of the bases and the height

On the other hand, the area of ​​a trapezoid is equal to the sum of the areas of the resulting triangles:

Equating these expressions, we get:

or with 2 = a 2 + b 2

4.3 Oldestproof(containedVonefromworksBhaskars).

Let ABCD be a square whose side is equal to the hypotenuse of the right triangle ABE (AB = c, BE = a, AE = b);

Let CK BE = a, DL CK, AM DL

DABE = ?BCK = ?CDL = ?AMD,

means KL = LM = ME = EK = a-b.

4.4 Proofsimplest

This proof is obtained in the simplest case of an isosceles right triangle.

This is probably where the theorem began.

In fact, it is enough just to look at the mosaic of isosceles right triangles to be convinced of the validity of the theorem.

For example, for triangle ABC: the square built on the hypotenuse AC contains 4 original triangles, and the squares built on the sides contain two. The theorem has been proven.

4.5 ProofancientHindus[ 2]

A square with side (a+b) can be divided into parts either as in figure a) or as in figure b). It is clear that parts 1, 2, 3, 4 are the same in both pictures. And if you subtract equals from equal (areas), then they will remain equal, i.e. With 2 = A 2 + b 2 .

A) b)

However,ancientHindus,whichbelongsThisreasoning,usuallyNotrecordedhis,Aaccompaniedonlyonein a word:Look!

4.6 ProofEuclid

For two millennia, the most widely used proof of the Pythagorean theorem was that of Euclid. It is placed in his famous book “Principles”.

Euclid lowered the height BN from the vertex of the right angle to the hypotenuse and proved that its continuation divides the square completed on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the sides.

The drawing used to prove this theorem is jokingly called “Pythagorean pants.” For a long time it was considered one of the symbols of mathematical science.

Students of the Middle Ages considered the proof of the Pythagorean theorem very difficult and called it Dons asinorum - donkey bridge, or elefuga - flight of the “poor”, since some “poor” students who did not have serious mathematical training fled from geometry. Weak students who memorized the theorems without understanding them, and were therefore nicknamed “donkeys,” were unable to overcome the Pythagorean theorem, which served as an insurmountable bridge for them. Because of the drawings accompanying the Pythagorean theorem, students also called it a “windmill,” composed poems like “Pythagorean pants are equal on all sides,” and drew cartoons.

5. ApplicationtheoremsPythagoras

5.1 Taskstheoreticalmodern

1. The perimeter of a rhombus is 68 cm, and one of its diagonals is 30 cm. Find the length of the other diagonal of the rhombus.

The hypotenuse KR of the right triangle KMR is equal to cm, and the leg MR is equal to 4 cm. Find the median RS.

Squares are constructed on the sides of a right triangle, and

S 1 -S 2 = 112 cm 2, and S 3 = 400 cm 2. Find the perimeter of the triangle.

Given triangle ABC, angle C = 90 0, CD AB, AC = 15 cm, AD = 9 cm.

Find AB.

5.2 Taskspracticalvintage

To secure the mast you need to install 4 cables. One end of each cable should be attached at a height of 12 m, the other on the ground at a distance of 5 m from the mast. Is 50 m of cable enough to secure the mast?

TaskIndianmathematicsXIIcenturyBhaskars

A lonely poplar grew on the river bank

Suddenly a gust of wind broke its trunk.

The poor poplar fell. And the angle is right

With the flow of the river its trunk formed.

Remember now that there is a river in that place

It was only four feet wide.

The top leaned at the edge of the river.

There are only three feet left of the trunk,

I ask you, tell me soon:

How tall is the poplar?”

Taskfromtextbook"Arithmetic"LeontiaMagnitsky

“If a certain person happens to build a ladder up to a wall, the height of the wall is 117 feet. And you will find a ladder 125 feet long.

And he wants to know how many stops sowing the stairs to defend the lower end from the wall."

TaskfromChinese"MathematiciansVninebooks"

“There is a reservoir with a side of 1 zhang = 10 chi. In the center of it there is a reed that protrudes above the water by 1 chi. If you pull the reed towards the shore, it will just touch it.

The question is: what is the depth of the water and what is the length of the reeds?

Conclusion

The Pythagorean theorem is so famous that it is difficult to imagine a person who has not heard of it. We studied a number of historical and mathematical sources, including information on the Internet, and saw that the Pythagorean theorem is interesting not only for its history, but also because it occupies an important place in life and science. This is evidenced by the various interpretations of the text of this theorem and the ways of its proof given in this work.

So, the Pythagorean theorem is one of the main and, one might say, the most important theorem of geometry. Its significance lies in the fact that most of the theorems of geometry can be deduced from it or with its help. The Pythagorean theorem is also remarkable because in itself it is not at all obvious. For example, the properties of an isosceles triangle can be seen directly in the drawing. But no matter how much you look at a right triangle, you will never see that there is a simple relationship between its sides: c 2 =a 2 +b 2. Therefore, visualization is often used to prove it.

The merit of Pythagoras was that he gave a complete scientific proof this theorem.

The personality of the scientist himself, whose memory is not coincidentally preserved by this theorem, is interesting. Pythagoras is a wonderful speaker, teacher and educator, organizer of his school, focused on the harmony of music and numbers, goodness and justice, knowledge and healthy image life. He may well serve as an example for us, distant descendants.

LiteratureAndAndInternet resources:

G.I. Glazer History of mathematics in school grades VII-VIII, manual for teachers, - M: Prosveshchenie 1982.

AND I. Dempan, N.Ya. Vilenkin “Behind the pages of a mathematics textbook” A manual for students in grades 5-6, Moscow, Education 1989.

I.G. Zenkevich “Aesthetics of a mathematics lesson”, M.: Education 1981.

Voitikova N.V. "Pythagorean Theorem" course work, Anzhero-Sudzhensk, 1999

V. Litzman. Pythagorean Theorem, M. 1960.

A.V. Voloshinov “Pythagoras” M. 1993.

L.F. Pichurin “Behind the pages of an algebra textbook” M. 1990.

A.N. Zemlyakov “Geometry in 10th grade” M. 1986.

V.V. Afanasyev “Formation of creative activity of students in the process of solving mathematical problems” Yaroslavl 1996.

P.I. Altynov “Tests. Geometry 7-9 grades.” M. 1998.

Newspaper "Mathematics" 17/1996.

Newspaper "Mathematics" 3/1997.

N.P. Antonov, M.Ya. Vygodsky, V.V. Nikitin, A.I. Sankin “Collection of problems in elementary mathematics.” M. 1963.

G.V. Dorofeev, M.K. Potapov, N.Kh. Rozov "Mathematics Manual". M. 1973

A.I. Shchetnikov “Pythagorean doctrine of number and magnitude.” Novosibirsk 1997.

“Real numbers. Irrational expressions" 8th grade. Tomsk University Publishing House. Tomsk - 1997.

M.S. Atanasyan “Geometry” 7-9 grades. M: Enlightenment, 1991

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