Creation of the Pythagorean theorem. Famous theorems (Pythagorean theorem)

The potential for creativity is usually attributed to the humanities, leaving the natural science to analysis, a practical approach and the dry language of formulas and numbers. Mathematics cannot be classified as a humanities subject. But without creativity you won’t go far in the “queen of all sciences” - people have known this for a long time. Since the time of Pythagoras, for example.

School textbooks, unfortunately, usually do not explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and feel its fundamental principles. And at the same time, try to free your mind from cliches and elementary truths - only in such conditions are all great discoveries born.

Such discoveries include what we know today as the Pythagorean theorem. With its help, we will try to show that mathematics not only can, but should be exciting. And that this adventure is suitable not only for nerds with thick glasses, but for everyone who is strong in mind and strong in spirit.

From the history of the issue

Strictly speaking, although the theorem is called the “Pythagorean theorem,” Pythagoras himself did not discover it. The right triangle and its special properties were studied long before it. There are two polar points of view on this issue. According to one version, Pythagoras was the first to find a complete proof of the theorem. According to another, the proof does not belong to the authorship of Pythagoras.

Today you can no longer check who is right and who is wrong. What is known is that the proof of Pythagoras, if it ever existed, has not survived. However, there are suggestions that the famous proof from Euclid’s Elements may belong to Pythagoras, and Euclid only recorded it.

It is also known today that problems about a right triangle are found in Egyptian sources from the time of Pharaoh Amenemhat I, on Babylonian clay tablets from the reign of King Hammurabi, in the ancient Indian treatise “Sulva Sutra” and the ancient Chinese work “Zhou-bi suan jin”.

As you can see, the Pythagorean theorem has occupied the minds of mathematicians since ancient times. This is confirmed by about 367 different pieces of evidence that exist today. In this, no other theorem can compete with it. Among the famous authors of proofs we can recall Leonardo da Vinci and the twentieth US President James Garfield. All this speaks of the extreme importance of this theorem for mathematics: most of the theorems of geometry are derived from it or are somehow connected with it.

Proofs of the Pythagorean theorem

School textbooks mostly give algebraic proofs. But the essence of the theorem is in geometry, so let’s first consider those proofs of the famous theorem that are based on this science.

Evidence 1

For the simplest proof of the Pythagorean theorem for a right triangle, you need to set ideal conditions: let the triangle be not only right-angled, but also isosceles. There is reason to believe that it was precisely this kind of triangle that ancient mathematicians initially considered.

Statement “a square built on the hypotenuse of a right triangle is equal to the sum of the squares built on its legs” can be illustrated with the following drawing:

Look at the isosceles right triangle ABC: On the hypotenuse AC, you can construct a square consisting of four triangles equal to the original ABC. And on sides AB and BC a square is built, each of which contains two similar triangles.

By the way, this drawing formed the basis of numerous jokes and cartoons dedicated to the Pythagorean theorem. The most famous is probably "Pythagorean pants are equal in all directions":

Evidence 2

This method combines algebra and geometry and can be considered a variant of the ancient Indian proof of the mathematician Bhaskari.

Construct a right triangle with sides a, b and c(Fig. 1). Then construct two squares with sides equal to the sum of the lengths of the two legs - (a+b). In each of the squares, make constructions as in Figures 2 and 3.

In the first square, build four triangles similar to those in Figure 1. The result is two squares: one with side a, the second with side b.

In the second square, four similar triangles constructed form a square with a side equal to the hypotenuse c.

The sum of the areas of the constructed squares in Fig. 2 is equal to the area of ​​the square we constructed with side c in Fig. 3. This can be easily checked by calculating the area of ​​the squares in Fig. 2 according to the formula. And the area of ​​the inscribed square in Figure 3. by subtracting the areas of four equal right triangles inscribed in the square from the area of ​​a large square with a side (a+b).

Writing all this down, we have: a 2 +b 2 =(a+b) 2 – 2ab. Open the brackets, carry out all the necessary algebraic calculations and get that a 2 +b 2 = a 2 +b 2. In this case, the area inscribed in Fig. 3. square can also be calculated using the traditional formula S=c 2. Those. a 2 +b 2 =c 2– you have proven the Pythagorean theorem.

Evidence 3

The ancient Indian proof itself was described in the 12th century in the treatise “The Crown of Knowledge” (“Siddhanta Shiromani”) and as the main argument the author uses an appeal addressed to the mathematical talents and observation skills of students and followers: “Look!”

But we will analyze this proof in more detail:

Inside the square, build four right triangles as indicated in the drawing. Let us denote the side of the large square, also known as the hypotenuse, With. Let's call the legs of the triangle A And b. According to the drawing, the side of the inner square is (a-b).

Use the formula for the area of ​​a square S=c 2 to calculate the area of ​​the outer square. And at the same time calculate the same value by adding the area of ​​the inner square and the areas of all four right triangles: (a-b) 2 2+4*1\2*a*b.

You can use both options for calculating the area of ​​a square to make sure that they give the same result. And this gives you the right to write down that c 2 =(a-b) 2 +4*1\2*a*b. As a result of the solution, you will receive the formula of the Pythagorean theorem c 2 =a 2 +b 2. The theorem has been proven.

Proof 4

This curious ancient Chinese proof was called the “Bride’s Chair” - because of the chair-like figure that results from all the constructions:

It uses the drawing that we have already seen in Fig. 3 in the second proof. And the inner square with side c is constructed in the same way as in the ancient Indian proof given above.

If you mentally cut off two green rectangular triangles from the drawing in Fig. 1, move them to opposite sides of the square with side c and attach the hypotenuses to the hypotenuses of the lilac triangles, you will get a figure called “bride’s chair” (Fig. 2). For clarity, you can do the same with paper squares and triangles. You will make sure that the “bride’s chair” is formed by two squares: small ones with a side b and big with a side a.

These constructions allowed the ancient Chinese mathematicians and us, following them, to come to the conclusion that c 2 =a 2 +b 2.

Evidence 5

This is another way to find a solution to the Pythagorean theorem using geometry. It's called the Garfield Method.

Construct a right triangle ABC. We need to prove that BC 2 = AC 2 + AB 2.

To do this, continue the leg AC and construct a segment CD, which is equal to the leg AB. Lower the perpendicular AD line segment ED. Segments ED And AC are equal. Connect the dots E And IN, and E And WITH and get a drawing like the picture below:

To prove the tower, we again resort to the method we have already tried: we find the area of ​​the resulting figure in two ways and equate the expressions to each other.

Find the area of ​​a polygon ABED can be done by adding up the areas of the three triangles that form it. And one of them, ERU, is not only rectangular, but also isosceles. Let's also not forget that AB=CD, AC=ED And BC=SE– this will allow us to simplify the recording and not overload it. So, S ABED =2*1/2(AB*AC)+1/2ВС 2.

At the same time, it is obvious that ABED- This is a trapezoid. Therefore, we calculate its area using the formula: S ABED =(DE+AB)*1/2AD. For our calculations, it is more convenient and clearer to represent the segment AD as the sum of segments AC And CD.

Let's write down both ways to calculate the area of ​​a figure, putting an equal sign between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). We use the equality of segments already known to us and described above to simplify the right side of the notation: AB*AC+1/2BC 2 =1/2(AB+AC) 2. Now let’s open the brackets and transform the equality: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having completed all the transformations, we get exactly what we need: BC 2 = AC 2 + AB 2. We have proven the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be proven using vectors, complex numbers, differential equations, stereometry, etc. And even physicists: if, for example, liquid is poured into square and triangular volumes similar to those shown in the drawings. By pouring liquid, you can prove the equality of areas and the theorem itself as a result.

A few words about Pythagorean triplets

This issue is little or not studied at all in the school curriculum. Meanwhile, he is very interesting and has great importance in geometry. Pythagorean triples are used to solve many mathematical problems. Understanding them may be useful to you in further education.

So what are Pythagorean triplets? This is the name for natural numbers collected in groups of three, the sum of the squares of two of which is equal to the third number squared.

Pythagorean triples can be:

• primitive (all three numbers are relatively prime);
• not primitive (if each number of a triple is multiplied by the same number, you get a new triple, which is not primitive).

Even before our era, the ancient Egyptians were fascinated by the mania for numbers of Pythagorean triplets: in problems they considered a right triangle with sides of 3, 4 and 5 units. By the way, any triangle whose sides are equal to the numbers from the Pythagorean triple is rectangular by default.

Examples of Pythagorean triplets: (3, 4, 5), (6, 8, 10), (5, 12, 13), (9, 12, 15), (8, 15, 17), (12, 16, 20 ), (15, 20, 25), (7, 24, 25), (10, 24, 26), (20, 21, 29), (18, 24, 30), (10, 30, 34), (21, 28, 35), (12, 35, 37), (15, 36, 39), (24, 32, 40), (9, 40, 41), (27, 36, 45), (14 , 48, 50), (30, 40, 50), etc.

Practical application of the theorem

The Pythagorean theorem is used not only in mathematics, but also in architecture and construction, astronomy and even literature.

First, about construction: the Pythagorean theorem is widely used in problems of various levels of complexity. For example, look at a Romanesque window:

Let us denote the width of the window as b, then the radius of the major semicircle can be denoted as R and express through b: R=b/2. The radius of smaller semicircles can also be expressed through b: r=b/4. In this problem we are interested in the radius of the inner circle of the window (let's call it p).

The Pythagorean theorem is just useful to calculate R. To do this, we use a right triangle, which is indicated by a dotted line in the figure. The hypotenuse of a triangle consists of two radii: b/4+p. One leg represents the radius b/4, another b/2-p. Using the Pythagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Next, we open the brackets and get b 2 /16+ bp/2+p 2 =b 2 /16+b 2 /4-bp+p 2. Let's transform this expression into bp/2=b 2 /4-bp. And then we divide all terms by b, we present similar ones to get 3/2*p=b/4. And in the end we find that p=b/6- which is what we needed.

Using the theorem, you can calculate the length of the rafters for gable roof. Determine how high a mobile communications tower is needed for the signal to reach a certain populated area. And even install steadily christmas tree on the city square. As you can see, this theorem lives not only on the pages of textbooks, but is often useful in real life.

In literature, the Pythagorean theorem has inspired writers since antiquity and continues to do so in our time. For example, the nineteenth-century German writer Adelbert von Chamisso was inspired to write a sonnet:

The light of truth will not dissipate soon,
But, having shone, it is unlikely to dissipate
And, like thousands of years ago,
It will not cause doubt or controversy.

The wisest when it touches your gaze
Light of truth, thank the gods;
And a hundred bulls, slaughtered, lie -
A return gift from the lucky Pythagoras.

Since then the bulls have been roaring desperately:
Forever alarmed the bull tribe
Event mentioned here.

It seems to them that the time is about to come,
And they will be sacrificed again
Some great theorem.

(translation by Viktor Toporov)

And in the twentieth century, the Soviet writer Evgeny Veltistov, in his book “The Adventures of Electronics,” devoted an entire chapter to proofs of the Pythagorean theorem. And another half chapter to the story about the two-dimensional world that could exist if the Pythagorean theorem became a fundamental law and even a religion for a single world. Living there would be much easier, but also much more boring: for example, no one there understands the meaning of the words “round” and “fluffy”.

And in the book “The Adventures of Electronics,” the author, through the mouth of mathematics teacher Taratar, says: “The main thing in mathematics is the movement of thought, new ideas.” It is precisely this creative flight of thought that gives rise to the Pythagorean theorem - it is not for nothing that it has so many varied proofs. It helps you go beyond the boundaries of the familiar and look at familiar things in a new way.

Conclusion

This article was created so that you can look beyond the school curriculum in mathematics and learn not only those proofs of the Pythagorean theorem that are given in the textbooks “Geometry 7-9” (L.S. Atanasyan, V.N. Rudenko) and “Geometry 7” -11” (A.V. Pogorelov), but also other interesting ways to prove the famous theorem. And also see examples of how the Pythagorean theorem can be applied in everyday life.

Firstly, this information will allow you to qualify for higher scores in mathematics lessons - information on the subject from additional sources is always highly appreciated.

Secondly, we wanted to help you feel how interesting mathematics is. Confirm with specific examples that there is always room for creativity. We hope that the Pythagorean theorem and this article will inspire you to independently explore and make exciting discoveries in mathematics and other sciences.

Tell us in the comments if you found the evidence presented in the article interesting. Did you find this information useful in your studies? Write to us what you think about the Pythagorean theorem and this article - we will be happy to discuss all this with you.

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One thing you can be one hundred percent sure of is that when asked what the square of the hypotenuse is, any adult will boldly answer: “The sum of the squares of the legs.” This theorem is firmly ingrained in the minds of every educated person, but you just need to ask someone to prove it, and difficulties can arise. So let's remember and consider different ways proof of the Pythagorean theorem.

Brief biography

The Pythagorean theorem is familiar to almost everyone, but for some reason the biography of the person who brought it into the world is not so popular. This can be fixed. Therefore, before exploring the different ways to prove Pythagoras’ theorem, you need to briefly get to know his personality.

Pythagoras - philosopher, mathematician, thinker originally from Today it is very difficult to distinguish his biography from the legends that have developed in memory of this great man. But as follows from the works of his followers, Pythagoras of Samos was born on the island of Samos. His father was an ordinary stone cutter, but his mother came from a noble family.

Judging by the legend, the birth of Pythagoras was predicted by a woman named Pythia, in whose honor the boy was named. According to her prediction, the born boy was supposed to bring a lot of benefit and good to humanity. Which is exactly what he did.

Birth of the theorem

In his youth, Pythagoras moved to Egypt to meet famous Egyptian sages there. After meeting with them, he was allowed to study, where he learned all the great achievements of Egyptian philosophy, mathematics and medicine.

It was probably in Egypt that Pythagoras was inspired by the majesty and beauty of the pyramids and created his great theory. This may shock readers, but modern historians believe that Pythagoras did not prove his theory. But he only passed on his knowledge to his followers, who later completed all the necessary mathematical calculations.

Be that as it may, today not one method of proving this theorem is known, but several at once. Today we can only guess how exactly the ancient Greeks performed their calculations, so here we will look at different ways to prove the Pythagorean theorem.

Pythagorean theorem

Before you begin any calculations, you need to figure out what theory you want to prove. The Pythagorean theorem goes like this: “In a triangle in which one of the angles is 90°, the sum of the squares of the legs is equal to the square of the hypotenuse.”

There are a total of 15 different ways to prove the Pythagorean theorem. This is a fairly large number, so we will pay attention to the most popular of them.

Method one

First, let's define what we've been given. These data will also apply to other methods of proving the Pythagorean theorem, so it is worth immediately remembering all the available notations.

Suppose we are given a right triangle with legs a, b and a hypotenuse equal to c. The first method of proof is based on the fact that you need to draw a square from a right triangle.

To do this, you need to add a segment equal to leg b to the leg of length a, and vice versa. This should result in two equal sides of the square. All that remains is to draw two parallel lines, and the square is ready.

Inside the resulting figure, you need to draw another square with a side equal to the hypotenuse of the original triangle. To do this, from the vertices ас and св you need to draw two parallel segments equal to с. Thus, we get three sides of the square, one of which is the hypotenuse of the original right triangle. All that remains is to draw the fourth segment.

Based on the resulting figure, we can conclude that the area of ​​the outer square is (a + b) 2. If you look inside the figure, you can see that in addition to the inner square, there are four right triangles. The area of ​​each is 0.5av.

Therefore, the area is equal to: 4 * 0.5ab + c 2 = 2av + c 2

Hence (a+c) 2 =2ab+c 2

And, therefore, c 2 =a 2 +b 2

The theorem has been proven.

Method two: similar triangles

This formula for proving the Pythagorean theorem was derived based on a statement from the section of geometry about similar triangles. It states that the leg of a right triangle is the average proportional to its hypotenuse and the segment of the hypotenuse emanating from the vertex of the 90° angle.

The initial data remains the same, so let's start right away with the proof. Let us draw a segment CD perpendicular to side AB. Based on the above statement, the sides of the triangles are equal:

AC=√AB*AD, SV=√AB*DV.

To answer the question of how to prove the Pythagorean theorem, the proof must be completed by squaring both inequalities.

AC 2 = AB * AD and CB 2 = AB * DV

Now we need to add up the resulting inequalities.

AC 2 + CB 2 = AB * (AD * DV), where AD + DV = AB

It turns out that:

AC 2 + CB 2 =AB*AB

And therefore:

AC 2 + CB 2 = AB 2

Proof of the Pythagorean theorem and various ways its solutions require a multifaceted approach to this problem. However, this option is one of the simplest.

Another calculation method

Descriptions of different methods of proving the Pythagorean theorem may not mean anything until you start practicing on your own. Many techniques involve not only mathematical calculations, but also the construction of new figures from the original triangle.

In this case, it is necessary to complete another right triangle VSD from the side BC. Thus, now there are two triangles with a common leg BC.

Knowing that the areas of similar figures have a ratio as the squares of their similar linear dimensions, then:

S avs * c 2 - S avd * in 2 = S avd * a 2 - S vsd * a 2

S avs *(from 2 - to 2) = a 2 *(S avd -S vsd)

from 2 - to 2 =a 2

c 2 =a 2 +b 2

Since out of the various methods of proving the Pythagorean theorem for grade 8, this option is hardly suitable, you can use the following method.

The easiest way to prove the Pythagorean Theorem. Reviews

According to historians, this method was first used to prove the theorem back in ancient Greece. It is the simplest, as it does not require absolutely any calculations. If you draw the picture correctly, then the proof of the statement that a 2 + b 2 = c 2 will be clearly visible.

The conditions for this method will be slightly different from the previous one. To prove the theorem, assume that right triangle ABC is isosceles.

We take the hypotenuse AC as the side of the square and draw its three sides. In addition, it is necessary to draw two diagonal lines in the resulting square. So that inside it you get four isosceles triangles.

You also need to draw a square to the legs AB and CB and draw one diagonal straight line in each of them. We draw the first line from vertex A, the second from C.

Now you need to carefully look at the resulting drawing. Since on the hypotenuse AC there are four triangles equal to the original one, and on the sides there are two, this indicates the veracity of this theorem.

By the way, thanks to this method of proving the Pythagorean theorem, the famous phrase was born: “Pythagorean pants are equal in all directions.”

Proof by J. Garfield

James Garfield is the twentieth President of the United States of America. In addition to making his mark on history as the ruler of the United States, he was also a gifted autodidact.

At the beginning of his career he was an ordinary teacher in a public school, but soon became the director of one of the highest educational institutions. The desire for self-development allowed him to propose a new theory for proving the Pythagorean theorem. The theorem and an example of its solution are as follows.

First you need to draw two right triangles on a piece of paper so that the leg of one of them is a continuation of the second. The vertices of these triangles need to be connected to ultimately form a trapezoid.

As you know, the area of ​​a trapezoid is equal to the product of half the sum of its bases and its height.

S=a+b/2 * (a+b)

If we consider the resulting trapezoid as a figure consisting of three triangles, then its area can be found as follows:

S=av/2 *2 + s 2 /2

Now we need to equalize the two original expressions

2ab/2 + c/2=(a+b) 2 /2

c 2 =a 2 +b 2

More than one volume could be written about the Pythagorean theorem and methods of proving it. teaching aid. But is there any point in it when this knowledge cannot be applied in practice?

Practical application of the Pythagorean theorem

Unfortunately, in modern school programs This theorem is intended to be used only in geometric problems. Graduates will soon leave school without knowing how they can apply their knowledge and skills in practice.

In fact, anyone can use the Pythagorean theorem in their daily life. And not only in professional activity, but also in ordinary household chores. Let's consider several cases when the Pythagorean theorem and methods of proving it may be extremely necessary.

Relationship between the theorem and astronomy

It would seem how stars and triangles on paper can be connected. In fact, astronomy is a scientific field in which the Pythagorean theorem is widely used.

For example, consider the movement of a light beam in space. It is known that light moves in both directions at the same speed. Let's call the trajectory AB along which the light ray moves l. And let's call half the time it takes light to get from point A to point B t. And the speed of the beam - c. It turns out that: c*t=l

If you look at this same ray from another plane, for example, from a space liner that moves with speed v, then when observing bodies in this way, their speed will change. In this case, even stationary elements will begin to move with speed v in the opposite direction.

Let's say the comic liner is sailing to the right. Then points A and B, between which the beam rushes, will begin to move to the left. Moreover, when the beam moves from point A to point B, point A has time to move and, accordingly, the light will already arrive at a new point C. To find half the distance by which point A has moved, you need to multiply the speed of the liner by half the travel time of the beam (t ").

And to find how far a ray of light could travel during this time, you need to mark half the path with a new letter s and get the following expression:

If we imagine that points of light C and B, as well as the space liner, are the vertices of an isosceles triangle, then the segment from point A to the liner will divide it into two right triangles. Therefore, thanks to the Pythagorean theorem, you can find the distance that a ray of light could travel.

This example, of course, is not the most successful, since only a few can be lucky enough to try it in practice. Therefore, let's consider more mundane applications of this theorem.

Mobile signal transmission range

Modern life can no longer be imagined without the existence of smartphones. But how much use would they be if they couldn’t connect subscribers via mobile communications?!

The quality of mobile communications directly depends on the height at which the mobile operator’s antenna is located. In order to calculate how far from a mobile tower a phone can receive a signal, you can apply the Pythagorean theorem.

Let's say you need to find the approximate height of a stationary tower so that it can distribute a signal within a radius of 200 kilometers.

AB (tower height) = x;

BC (signal transmission radius) = 200 km;

OS (radius of the globe) = 6380 km;

OB=OA+ABOB=r+x

Applying the Pythagorean theorem, we find out that the minimum height of the tower should be 2.3 kilometers.

Pythagorean theorem in everyday life

Oddly enough, the Pythagorean theorem can be useful even in everyday matters, such as determining the height of a wardrobe, for example. At first glance, there is no need to use such complex calculations, because you can simply take measurements using a tape measure. But many people wonder why certain problems arise during the assembly process if all measurements were taken more than accurately.

The fact is that the wardrobe is assembled in a horizontal position and only then raised and installed against the wall. Therefore, during the process of lifting the structure, the side of the cabinet must move freely both along the height and diagonally of the room.

Let's assume there is a wardrobe with a depth of 800 mm. Distance from floor to ceiling - 2600 mm. An experienced furniture maker will say that the height of the cabinet should be 126 mm less than the height of the room. But why exactly 126 mm? Let's look at an example.

With ideal cabinet dimensions, let’s check the operation of the Pythagorean theorem:

AC =√AB 2 +√BC 2

AC=√2474 2 +800 2 =2600 mm - everything fits.

Let's say the height of the cabinet is not 2474 mm, but 2505 mm. Then:

AC=√2505 2 +√800 2 =2629 mm.

Therefore, this cabinet is not suitable for installation in this room. Because lifting it into a vertical position can cause damage to its body.

Perhaps, having considered different ways of proving the Pythagorean theorem by different scientists, we can conclude that it is more than true. Now you can use the information received in your daily life and be completely confident that all calculations will be not only useful, but also correct.

Pythagorean theorem: Sum of areas of squares resting on legs ( a And b), equal to the area of ​​the square built on the hypotenuse ( c).

Geometric formulation:

The theorem was originally formulated as follows:

Algebraic formulation:

That is, denoting the length of the hypotenuse of the triangle by c, and the lengths of the legs through a And b :

a 2 + b 2 = c 2

Both formulations of the theorem are equivalent, but the second formulation is more elementary; it does not require the concept of area. That is, the second statement can be verified without knowing anything about the area and by measuring only the lengths of the sides of a right triangle.

Converse Pythagorean theorem:

Proof

At the moment, 367 proofs of this theorem have been recorded in the scientific literature. Probably, the Pythagorean theorem is the only theorem with such an impressive number of proofs. Such diversity can only be explained by the fundamental significance of the theorem for geometry.

Of course, conceptually all of them can be divided into a small number of classes. The most famous of them: proofs by the area method, axiomatic and exotic proofs (for example, using differential equations).

Through similar triangles

The following proof of the algebraic formulation is the simplest of the proofs, constructed directly from the axioms. In particular, it does not use the concept of area of ​​a figure.

Let ABC there is a right triangle with a right angle C. Let's draw the height from C and denote its base by H. Triangle ACH similar to a triangle ABC at two corners. Likewise, triangle CBH similar ABC. By introducing the notation

we get

What is equivalent

Adding it up, we get

Proofs using the area method

The proofs below, despite their apparent simplicity, are not so simple at all. They all use properties of area, the proof of which is more complex than the proof of the Pythagorean theorem itself.

Proof via equicomplementation

1. Let's arrange four equal right triangles as shown in Figure 1.
2. Quadrangle with sides c is a square, since the sum of two acute angles is 90°, and the straight angle is 180°.
3. The area of ​​the entire figure is equal, on the one hand, to the area of ​​a square with side (a + b), and on the other hand, to the sum of the areas of four triangles and two internal squares.

Q.E.D.

Proofs through equivalence

Elegant proof using permutation

An example of one such proof is shown in the drawing on the right, where a square built on the hypotenuse is rearranged into two squares built on the legs.

Euclid's proof

Drawing for Euclid's proof

Illustration for Euclid's proof

The idea of ​​Euclid's proof is as follows: let's try to prove that half the area of ​​the square built on the hypotenuse is equal to the sum of the half areas of the squares built on the legs, and then the areas of the large and two small squares are equal.

Let's look at the drawing on the left. On it we constructed squares on the sides of a right triangle and drew a ray s from the vertex of the right angle C perpendicular to the hypotenuse AB, it cuts the square ABIK, built on the hypotenuse, into two rectangles - BHJI and HAKJ, respectively. It turns out that the areas of these rectangles are exactly equal to the areas of the squares built on the corresponding legs.

Let's try to prove that the area of ​​the square DECA is equal to the area of ​​the rectangle AHJK. To do this, we will use an auxiliary observation: The area of ​​a triangle with the same height and base as the given rectangle is equal to half the area of ​​the given rectangle. This is a consequence of defining the area of ​​a triangle as half the product of the base and the height. From this observation it follows that the area of ​​triangle ACK is equal to the area of ​​triangle AHK (not shown in the figure), which in turn is equal to half the area of ​​rectangle AHJK.

Let us now prove that the area of ​​triangle ACK is also equal to half the area of ​​square DECA. The only thing that needs to be done for this is to prove the equality of triangles ACK and BDA (since the area of ​​triangle BDA is equal to half the area of ​​the square according to the above property). The equality is obvious, the triangles are equal on both sides and the angle between them. Namely - AB=AK,AD=AC - the equality of the angles CAK and BAD is easy to prove by the method of motion: we rotate the triangle CAK 90° counterclockwise, then it is obvious that the corresponding sides of the two triangles in question will coincide (due to the fact that the angle at the vertex of the square is 90°).

The reasoning for the equality of the areas of the square BCFG and the rectangle BHJI is completely similar.

Thus, we proved that the area of ​​a square built on the hypotenuse is composed of the areas of squares built on the legs. The idea behind this proof is further illustrated by the animation above.

Proof of Leonardo da Vinci

Proof of Leonardo da Vinci

The main elements of the proof are symmetry and motion.

Let's consider the drawing, as can be seen from the symmetry, a segment CI cuts the square ABHJ into two identical parts (since triangles ABC And JHI equal in construction). Using a 90 degree counterclockwise rotation, we see the equality of the shaded figures CAJI And GDAB . Now it is clear that the area of ​​the figure we have shaded is equal to the sum of half the areas of the squares built on the legs and the area of ​​the original triangle. On the other hand, it is equal to half the area of ​​the square built on the hypotenuse, plus the area of ​​the original triangle. The last step in the proof is left to the reader.

Proof by the infinitesimal method

The following proof using differential equations is often attributed to the famous English mathematician Hardy, who lived in the first half of the 20th century.

Looking at the drawing shown in the figure and observing the change in side a, we can write the following relation for infinitesimal side increments With And a(using triangle similarity):

Proof by the infinitesimal method

Using the method of separation of variables, we find

More general expression to change the hypotenuse in case of increments of both legs

Integrating this equation and using the initial conditions, we obtain

c 2 = a 2 + b 2 + constant.

Thus we arrive at the desired answer

c 2 = a 2 + b 2 .

As is easy to see, 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.

A simpler proof can be obtained if we assume that one of the legs does not experience an increment (in this case, the leg b). Then for the integration constant we obtain

Variations and generalizations

• If instead of squares we construct other similar figures on the sides, then the following generalization of the Pythagorean theorem is true: In a right triangle, the sum of the areas of similar figures built on the sides is equal to the area of ​​the figure built on the hypotenuse. In particular:
  • The sum of the areas of regular triangles built on the legs is equal to the area of ​​a regular triangle built on the hypotenuse.
  • The sum of the areas of semicircles built on the legs (as on the diameter) is equal to the area of ​​the semicircle built on the hypotenuse. This example is used to prove the properties of figures bounded by the arcs of two circles and called Hippocratic lunulae.

Story

Chu-pei 500–200 BC. On the left is the inscription: the sum of the squares of the lengths of the height and base is the square of the length of the hypotenuse.

The ancient Chinese book Chu-pei talks about a Pythagorean triangle with sides 3, 4 and 5: The same book offers a drawing that coincides with one of the drawings of the Hindu geometry of Bashara.

Cantor (the greatest German historian of mathematics) believes that the equality 3² + 4² = 5² was already known to the Egyptians around 2300 BC. e., during the time of King Amenemhat 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.

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. It could be objected to the Harpedonaptians that their method of construction becomes superfluous 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 about the Pythagorean theorem among the Babylonians. In one text dating back to the time of Hammurabi, that is, to 2000 BC. e., 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. Based, on the one hand, on the current level of knowledge about Egyptian and Babylonian mathematics, and on the other, on a critical study of Greek sources, Van der Waerden (Dutch mathematician) came to the following conclusion:

Literature

In Russian

• Skopets Z. A. Geometric miniatures. M., 1990
• Elensky Shch. In the footsteps of Pythagoras. M., 1961
• Van der Waerden B. L. Awakening Science. Mathematics of Ancient Egypt, Babylon and Greece. M., 1959
• Glazer G.I. History of mathematics at school. M., 1982
• W. Litzman, “The Pythagorean Theorem” M., 1960.
  • A site about the Pythagorean theorem with a large number of proofs, material taken from the book by V. Litzmann, a large number of drawings are presented in the form of separate graphic files.
• The Pythagorean theorem and Pythagorean triples chapter from the book by D. V. Anosov “A look at mathematics and something from it”
• About the Pythagorean theorem and methods of proving it G. Glaser, academician of the Russian Academy of Education, Moscow

In English

• Pythagorean Theorem at WolframMathWorld
• Cut-The-Knot, section on the Pythagorean theorem, about 70 proofs and extensive additional information (English)

Wikimedia Foundation. 2010.

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 isis 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 Metapontum, 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 amount 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 meant only 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 becamebasis of application numerical methods 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 stubbornly 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 ratios 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. Also 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.

Let's start our historical review 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 general form theorems.

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 strip 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 already 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 it 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 it a “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; line 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 yourselfequality 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.” Duringfor 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.

The great discoveries of Pythagoras the mathematician found their application in different times and all over the world. This applies to the greatest extent to the Pythagorean theorem.

For example, in China Special attention in this regard, one should pay attention to the mathematical book Chu-pei, which says this about the famous Pythagorean triangle, which has sides 3, 4, 5: “If you decompose a right angle into its component parts, then the line connecting the ends of all its sides will be 5, then as the base will be 3 and the height 4.” The same book shows a drawing that is similar to one of the drawings in the Hindu geometry of Bashara.

The outstanding German researcher of the history of mathematics Cantor believes that the Pythagorean equality 3? + 4? = 5? already known in Egypt around 2300 BC. BC, during the reign of King Amenemhat I (according to papyrus 6619 of the Berlin Museum). According to Kantor, the harpedonaptes, or the so-called “rope pullers,” built right angles using right triangles, the sides of which were 3, 4, 5. Their construction method is quite easily reproduced. If you take a piece of rope 12 m long, tie colored strips to it - one at a three-meter distance from one end, and the other 4 meters from the other, then a right angle will be enclosed between the two sides - 3 and 4 meters. One can object to the harpedonapts that this method of construction would be superfluous if we take, for example, the wooden triangle that all carpenters use. Indeed, there are Egyptian drawings, for example, depicting a carpenter's workshop, in which such a tool is found. But nevertheless, the fact remains that the Pythagorean triangle was used in ancient Egypt.

Little more information is available about the Pythagorean theorem used by the Babylonians. In the found text, which dates back to the time of Hammurabi, which is 2000 BC. e., there is an approximate definition of the hypotenuse of a right triangle. Consequently, this confirms that calculations with the sides of right triangles were already carried out in Mesopotamia, at least in some cases. Mathematician Van der Waerden from Holland, on the one hand, using the current level of knowledge about Babylonian and Egyptian mathematics, and on the other, based on a careful study of Greek sources, came to the following conclusions: “The merit of the first Greek mathematicians: Thales, Pythagoras and the Pythagoreans – not the discovery of mathematics, but its justification and systematization. They were able to turn computational recipes based on vague ideas into an exact science.”

Among the Hindus, along with the Babylonians and Egyptians, geometry was closely associated with cult. It is quite possible that the Pythagorean theorem was known in India already in the 18th century BC. e.

The "List of Mathematicians" supposedly compiled by Eudemus says of Pythagoras: "Pythagoras is reported to have turned the study of this branch of knowledge (geometry) into a real science, having analyzed its foundations from the highest point of view and examined its theories in a more mental and less material manner." .