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 entrenched in the minds of everyone. educated person, but all you have to do is ask someone to prove it, and difficulties may 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 carried out their calculations, so here we will look at different ways to prove the Pythagorean theorem.
Pythagorean theorem
Before you start 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 leg length a, and vice versa. This should make two equal sides 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 is 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
A description of different ways to prove the Pythagorean theorem may not mean anything until you start practicing it yourself. 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.”
J. Garfield's proof
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 offer new theory proof of 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, modern school curricula provide for the use of this theorem only in geometric problems. Graduates will soon leave school without knowing how they can apply their knowledge and skills in practice.
In fact, use the Pythagorean theorem in your everyday life everyone can. And not only in professional activities, 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 scientific field, which makes extensive use of the Pythagorean theorem.
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 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 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 would they be of much use if they could not 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.
The history of the Pythagorean theorem goes back several thousand years. A statement that states that it was known long before the birth of the Greek mathematician. However, the Pythagorean theorem, the history of its creation and its proof are associated for the majority with this scientist. According to some sources, the reason for this was the first proof of the theorem, which was given by Pythagoras. However, some researchers deny this fact.
Music and logic
Before telling how the history of the Pythagorean theorem developed, let us briefly look at the biography of the mathematician. He lived in the 6th century BC. The date of birth of Pythagoras is considered to be 570 BC. e., the place is the island of Samos. Little is known reliably about the life of the scientist. Biographical data in ancient Greek sources is intertwined with obvious fiction. On the pages of the treatises, he appears as a great sage with excellent command of words and the ability to persuade. By the way, this is why the Greek mathematician was nicknamed Pythagoras, that is, “persuasive speech.” According to another version, the birth of the future sage was predicted by Pythia. The father named the boy Pythagoras in her honor.
The sage learned from the great minds of the time. Among the teachers of the young Pythagoras are Hermodamantus and Pherecydes of Syros. The first instilled in him a love of music, the second taught him philosophy. Both of these sciences will remain the focus of the scientist throughout his life.
30 years of training
According to one version, being an inquisitive young man, Pythagoras left his homeland. He went to seek knowledge in Egypt, where he stayed, according to different sources, from 11 to 22 years old, and then was captured and sent to Babylon. Pythagoras was able to benefit from his position. For 12 years he studied mathematics, geometry and magic in the ancient state. Pythagoras returned to Samos only at the age of 56. The tyrant Polycrates ruled here at that time. Pythagoras could not accept such a political system and soon went to the south of Italy, where the Greek colony of Croton was located.
Today it is impossible to say for sure whether Pythagoras was in Egypt and Babylon. He may have left Samos later and went straight to Croton.
Pythagoreans
The history of the Pythagorean theorem is connected with the development of the school created by the Greek philosopher. This religious and ethical brotherhood preached observance of a special way of life, studied arithmetic, geometry and astronomy, and was engaged in the study of the philosophical and mystical side of numbers.
All the discoveries of the students of the Greek mathematician were attributed to him. However, the history of the emergence of the Pythagorean theorem is associated by ancient biographers only with the philosopher himself. It is assumed that he passed on to the Greeks the knowledge gained in Babylon and Egypt. There is also a version that he actually discovered the theorem on the relationship between the legs and the hypotenuse, without knowing about the achievements of other peoples.
Pythagorean theorem: history of discovery
Some ancient Greek sources describe Pythagoras' joy when he succeeded in proving the theorem. In honor of this event, he ordered a sacrifice to the gods in the form of hundreds of bulls and held a feast. Some scientists, however, point out the impossibility of such an act due to the peculiarities of the views of the Pythagoreans.
It is believed that in the treatise “Elements”, created by Euclid, the author provides a proof of the theorem, the author of which was the great Greek mathematician. However, not everyone supported this point of view. Thus, even the ancient Neoplatonist philosopher Proclus pointed out that the author of the proof given in the Elements was Euclid himself.
Be that as it may, the first person to formulate the theorem was not Pythagoras.
Ancient Egypt and Babylon
The Pythagorean theorem, the history of which is discussed in the article, according to the German mathematician Cantor, was known back in 2300 BC. e. in Egypt. The ancient inhabitants of the Nile Valley during the reign of Pharaoh Amenemhat I knew the equality 3 2 + 4 ² = 5 ². It is assumed that with the help of triangles with sides 3, 4 and 5, the Egyptian “rope pullers” built right angles.
They also knew the Pythagorean theorem in Babylon. On clay tablets dating back to 2000 BC. and dating back to the reign, an approximate calculation of the hypotenuse of a right triangle was discovered.
India and China
The history of the Pythagorean theorem is also connected with the ancient civilizations of India and China. The treatise “Zhou-bi suan jin” contains indications that (its sides are related as 3:4:5) was known in China back in the 12th century. BC e., and by the 6th century. BC e. mathematicians of this state knew general view theorems.
The construction of a right angle using the Egyptian triangle was also outlined in the Indian treatise “Sulva Sutra”, dating back to the 7th-5th centuries. BC e.
Thus, the history of the Pythagorean theorem by the time of the birth of the Greek mathematician and philosopher was already several hundred years old.
Proof
During its existence, the theorem became one of the fundamental ones in geometry. The history of the proof of the Pythagorean theorem probably began with the consideration of an equilateral square. Squares are constructed on its hypotenuse and legs. The one that “grew” on the hypotenuse will consist of four triangles equal to the first. The squares on the sides consist of two such triangles. A simple graphical representation clearly shows the validity of the statement formulated in the form of the famous theorem.
Another simple proof combines geometry with algebra. Four identical right triangles with sides a, b, c are drawn so that they form two squares: the outer one with side (a + b) and the inner one with side c. In this case, the area of the smaller square will be equal to c 2. The area of a large one is calculated from the sum of the areas small square and all triangles (the area of a right triangle, recall, is calculated by the formula (a * b) / 2), that is, c 2 + 4 * ((a * b) / 2), which is equal to c 2 + 2ab. The area of a large square can be calculated in another way - as the product of two sides, that is, (a + b) 2, which is equal to a 2 + 2ab + b 2. It turns out:
a 2 + 2ab + b 2 = c 2 + 2ab,
a 2 + b 2 = c 2.
There are many versions of the proof of this theorem. Euclid, Indian scientists, and Leonardo da Vinci worked on them. Often the ancient sages cited drawings, examples of which are located above, and did not accompany them with any explanations other than the note “Look!” The simplicity of the geometric proof, provided that some knowledge was available, did not require comments.
The history of the Pythagorean theorem, briefly outlined in the article, debunks the myth about its origin. However, it is difficult to even imagine that the name of the great Greek mathematician and philosopher will ever cease to be associated with it.
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 2Both 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
On 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
- Let's arrange four equal right triangles as shown in Figure 1.
- Quadrangle with sides c is a square, since the sum of two sharp corners 90°, and the unfolded angle is 180°.
- 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). This 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
A more general expression for the change in the hypotenuse in the case of increments on both sides
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 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.
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 hand, 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 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, large number 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.
The Pythagorean theorem is the most important statement of geometry. The theorem is formulated as follows: 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.
The discovery of this statement is usually attributed to the ancient Greek philosopher and mathematician Pythagoras (VI century BC). But a study of Babylonian cuneiform tablets and ancient Chinese manuscripts (copies of even older manuscripts) showed that this statement was known long before Pythagoras, perhaps a millennium before him. The merit of Pythagoras was that he discovered the proof of this theorem.
It is likely that the fact stated in the Pythagorean theorem was first established for isosceles right triangles. Just look at the mosaic of black and light triangles shown in Fig. 1, to verify the validity of the theorem for a triangle: a square built on the hypotenuse contains 4 triangles, and a square containing 2 triangles is built on each side. For proof general case V Ancient India placed in two ways: in a square with a side, they depicted four right triangles with legs of lengths and (Fig. 2, a and 2, b), after which they wrote one word “Look!” And indeed, looking at these drawings, we see that on the left there is a figure free of triangles, consisting of two squares with sides and, accordingly, its area is equal to , and on the right there is a square with a side - its area is equal to . This means that this constitutes the statement of the Pythagorean theorem.
However, for two thousand years, it was not this visual proof that was used, but a more complex proof invented by Euclid, which is placed in his famous book “Elements” (see Euclid and his “Elements”), Euclid lowered the height from the vertex of a right angle to the hypotenuse and proved , that its continuation divides the square built on the hypotenuse into two rectangles, the areas of which are equal to the areas of the corresponding squares built on the legs (Fig. 3). 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.
Today, several dozen different proofs of the Pythagorean theorem are known. Some of them are based on the partition of squares, in which a square built on the hypotenuse consists of parts included in the partitions of squares built on the legs; others - on the complement to equal figures; the third - on the fact that the height lowered from the vertex of a right angle to the hypotenuse divides a right triangle into two triangles similar to it.
The Pythagorean theorem underlies most geometric calculations. Even in Ancient Babylon, it was used to calculate the length of the height of an isosceles triangle from the lengths of the base and side, the arrow of a segment from the diameter of the circle and the length of the chord, and established the relationships between the elements of some regular polygons. Using the Pythagorean theorem, we prove its generalization, which allows us to calculate the length of the side lying opposite an acute or obtuse angle:
From this generalization it follows that the presence of a right angle in is not only sufficient, but also a necessary condition for the equality to be satisfied. From formula (1) follows the relation 
between the lengths of the diagonals and sides of a parallelogram, with the help of which it is easy to find the length of the median of a triangle from the lengths of its sides.
Based on the Pythagorean theorem, a formula is derived that expresses the area of any triangle through the lengths of its sides (see Heron's formula). Of course, the Pythagorean theorem was also used to solve various practical problems.
Instead of squares, you can build any similar figures (equilateral triangles, semicircles, etc.) on the sides of a right triangle. In this case, the area of the figure built on the hypotenuse is equal to the sum of the areas of the figures built on the legs. Another generalization is associated with the transition from plane to space. It is formulated as follows: the square of the diagonal length of a rectangular parallelepiped is equal to the sum of the squares of its dimensions (length, width and height). A similar theorem is true in multidimensional and even infinite-dimensional cases.
The Pythagorean theorem exists only in Euclidean geometry. It does not occur either in Lobachevsky geometry or in other non-Euclidean geometries. There is no analogue of the Pythagorean theorem on the sphere. Two meridians forming an angle of 90° and the equator bound on a sphere an equilateral spherical triangle, all three angles of which are right angles. For him, not as on a plane.
Using the Pythagorean theorem, calculate the distance between points and coordinate plane according to the formula
.
After the Pythagorean theorem was discovered, the question arose of how to find all triplets of natural numbers that can be sides of right triangles (see Fermat's last theorem). They were discovered by the Pythagoreans, but some general methods for finding such triplets of numbers were already known to the Babylonians. One of the cuneiform tablets contains 15 triplets. Among them there are triplets consisting of so many large numbers, that there can be no question of finding them by selection.
Hippocratic fossa
Hippocratic lunas are figures bounded by the arcs of two circles, and, moreover, such that using the radii and length of the common chord of these circles, using a compass and a ruler, one can construct squares of equal size to them.
From the generalization of the Pythagorean theorem to semicircles, it follows that the sum of the areas of the pink lumps shown in the figure on the left is equal to the area of the blue triangle. Therefore, if you take an isosceles right triangle, you will get two holes, the area of each of which will be equal to half the area of the triangle. Trying to solve the problem of squaring a circle (see Classical problems of antiquity), the ancient Greek mathematician Hippocrates (5th century BC) found several more holes, the areas of which are expressed in terms of the areas of rectilinear figures.
A complete list of hippomarginal lunulae was obtained only in the 19th-20th centuries. thanks to the use of Galois theory methods.
For those interested in the history of the Pythagorean theorem, which is studied in school curriculum, it will also be interesting to know such a fact as the publication in 1940 of a book with three hundred and seventy proofs of this seemingly simple theorem. But it intrigued the minds of many mathematicians and philosophers different eras. In the Guinness Book of Records it is recorded as the theorem with the maximum number of proofs.
History of Pythagorean Theorem
Associated with the name of Pythagoras, the theorem was known long before the birth of the great philosopher. Thus, in Egypt, during the construction of structures, the aspect ratio of a right triangle was taken into account five thousand years ago. Babylonian texts mention the same aspect ratio of a right triangle 1200 years before the birth of Pythagoras.
The question arises, why then does history say that the origin of the Pythagorean theorem belongs to him? There can be only one answer - he proved the ratio of sides in a triangle. He did what those who simply used the aspect ratio and hypotenuse established by experience did not do centuries ago.
From the life of Pythagoras
The future great scientist, mathematician, philosopher was born on the island of Samos in 570 BC. Historical documents have preserved information about the father of Pythagoras, who was a carver precious stones, but there is no information about the mother. They said about the boy who was born that he was an extraordinary child who showed childhood passion for music and poetry. Historians include Hermodamas and Pherecydes of Syros as the teachers of young Pythagoras. The first introduced the boy into the world of the muses, and the second, being a philosopher and founder of the Italian school of philosophy, directed the young man’s gaze to the logos.
At the age of 22 (548 BC), Pythagoras went to Naucratis to study the language and religion of the Egyptians. Next, his path lay in Memphis, where, thanks to the priests, having gone through their ingenious tests, he comprehended Egyptian geometry, which, perhaps, prompted the inquisitive young man to prove the Pythagorean theorem. History will later assign this name to the theorem.
Captivity of the King of Babylon
On his way home to Hellas, Pythagoras is captured by the king of Babylon. But being in captivity benefited the inquisitive mind of the aspiring mathematician; he had a lot to learn. Indeed, in those years mathematics in Babylon was more developed than in Egypt. He spent twelve years studying mathematics, geometry and magic. And, perhaps, it was Babylonian geometry that was involved in the proof of the ratio of the sides of a triangle and the history of the discovery of the theorem. Pythagoras had enough knowledge and time for this. But there is no documentary confirmation or refutation that this happened in Babylon.
In 530 BC. Pythagoras escapes from captivity to his homeland, where he lives at the court of the tyrant Polycrates in the status of a half-slave. Pythagoras is not satisfied with such a life, and he retires to the caves of Samos, and then goes to the south of Italy, where at that time the Greek colony of Croton was located.
Secret monastic order
On the basis of this colony, Pythagoras organized a secret monastic order, which was a religious union and a scientific society at the same time. This society had its own charter, which spoke about observing a special way of life.
Pythagoras argued that in order to understand God, a person must know such sciences as algebra and geometry, know astronomy and understand music. Research work boiled down to knowledge of the mystical side of numbers and philosophy. It should be noted that the principles preached at that time by Pythagoras make sense in imitation at the present time.
Many of the discoveries made by Pythagoras' students were attributed to him. However, in short, the history of the creation of the Pythagorean theorem by ancient historians and biographers of that time is directly associated with the name of this philosopher, thinker and mathematician.
Teachings of Pythagoras
Perhaps the idea of the connection between the theorem and the name of Pythagoras was prompted by the statement of the great Greek that all the phenomena of our life are encrypted in the notorious triangle with its legs and hypotenuse. And this triangle is the “key” to solving all emerging problems. The great philosopher said that you should see the triangle, then you can consider that the problem is two-thirds solved.
Pythagoras spoke about his teaching only to his students orally, without making any notes, keeping it secret. Unfortunately, the teachings of the greatest philosopher have not survived to this day. Something leaked out of it, but it is impossible to say how much is true and how much is false in what became known. Even with the history of the Pythagorean theorem, not everything is certain. Historians of mathematics doubt the authorship of Pythagoras; in their opinion, the theorem was used many centuries before his birth.
Pythagorean theorem
It may seem strange, but historical facts there is no proof of the theorem by Pythagoras himself - neither in the archives nor in any other sources. IN modern version it is believed to belong to none other than Euclid himself.
There is evidence from one of the greatest historians of mathematics, Moritz Cantor, who discovered on a papyrus stored in the Berlin Museum, written down by the Egyptians around 2300 BC. e. equality, which read: 3² + 4² = 5².
Brief history of the Pythagorean theorem
The formulation of the theorem from Euclidean “Principles”, in translation, sounds the same as in the modern interpretation. There is nothing new in her reading: the square of the opposite side right angle, is equal to the sum of the squares of the sides adjacent to the right angle. The fact that the ancient civilizations of India and China used the theorem is confirmed by the treatise “Zhou - bi suan jin”. It contains information about the Egyptian triangle, which describes the aspect ratio as 3:4:5.
No less interesting is another Chinese mathematical book, “Chu Pei,” which also mentions the Pythagorean triangle with explanations and drawings that coincide with the drawings of Hindu geometry by Bashara. About the triangle itself, the book says that if a right angle can be decomposed into its component parts, then the line that connects the ends of the sides will be equal to five if the base is equal to three and the height is equal to four.
Indian treatise "Sulva Sutra", dating back to approximately the 7th-5th centuries BC. e., talks about constructing a right angle using the Egyptian triangle.
Proof of the theorem
In the Middle Ages, students considered proving a theorem too difficult. Weak students learned theorems by heart, without understanding the meaning of the proof. In this regard, they received the nickname “donkeys”, because the Pythagorean theorem was an insurmountable obstacle for them, like a bridge for a donkey. In the Middle Ages, students came up with a humorous verse on the subject of this theorem.
To prove the Pythagorean theorem in the easiest way, you should simply measure its sides, without using the concept of areas in the proof. The length of the side opposite the right angle is c, and a and b adjacent to it, as a result we obtain the equation: a 2 + b 2 = c 2. This statement, as mentioned above, is verified by measuring the lengths of the sides of a right triangle.
If we begin the proof of the theorem by considering the area of the rectangles built on the sides of the triangle, we can determine the area of the entire figure. It will be equal to the area of a square with side (a + b), and on the other hand, the sum of the areas of four triangles and the inner square.
(a + b) 2 = 4 x ab/2 + c 2 ;
a 2 + 2ab + b 2 ;
c 2 = a 2 + b 2 , which is what needed to be proved.
The practical significance of the Pythagorean theorem is that it can be used to find the lengths of segments without measuring them. During the construction of structures, distances, placement of supports and beams are calculated, and centers of gravity are determined. The Pythagorean theorem applies in all modern technologies. They didn’t forget about the theorem when creating movies in 3D-6D dimensions, where in addition to the three dimensions we are used to: height, length, width, time, smell and taste are taken into account. How are tastes and smells related to the theorem, you ask? Everything is very simple - when showing a film, you need to calculate where and what smells and tastes to direct in the auditorium.
Maybe there will be more. Unlimited scope for discovering and creating new technologies awaits inquisitive minds.
