Various ways of proving the Pythagorean theorem: apply, describe and explain. How to fix the Pythagorean theorem Pythagoras

Pythagorean theorem- One of the main theorems of Euclidean geometry, which establishes special

between the sides of a rectangular tricot.

It is important that the Greek mathematician Pythagoras brought it to his honor, it is named after him.

Geometric formulation of the Pythagorean theorem.

Back on the cob, Boule's theorem is formulated in an offensive rank:

For a straight-cut tricot, the area of the square, pobudovanogo on the hypotenuse, more sum of the area of the squares,

wake up on the catheters.

Algebraic formulation of the Pythagorean theorem.

In a straight-cut tricot, the square of the length of the hypotenuse is more than the sum of the squares of the length of the legs.

Tobto, knowing the length of hypotension of the tricutnik through c, and dozhini cathetiv through aі b:

Offended formulary Pythagorean theorems equivalent, but to another, the formula is more elementary, it is not

I need to understand the area. That other firmness can be distorted, knowing nothing about that area

vimіryavshi only dozhini storіn pryamokutny trikutnik.

The return theorem of Pythagoras.

If the square of one side of the tricot is equal to the sum of the squares of the other two sides, then

tricutnik is rectilinear.

Abo, in other words:

For a triple of positive numbers a, bі c, such that

іsnuє rectocut tricutnik іz legs aі b that hypotenuse c.

The Pythagorean theorem for a equi-femoral tricot.

The Pythagorean theorem for a equilateral tricot.

Prove the Pythagorean theorem.

At the moment, 367 proofs of the theorem have been recorded in the scientific literature. Ymovirno, theorem

Pythagoras with a single theorem with a significant number of proofs. Take a raznomanittya

can be explained only by the fundamental meanings of the theorem for geometry.

Understanding, conceptually they can be divided into a small number of classes. Find some of them:

prove area method, axiomaticі exotic proof(for example,

for help differential ratios).

1. Proof of the Pythagorean theorem through tricks.

The coming proof of the algebraic formula is the simplest of proofs that will be

without intermediary with axioms. Zokrema won't be victorious in understanding the area of the figure.

Come on ABCє straight cut tricout with a straight cut C. Let's draw a height from C and significant

її zasnuvannya through H.

Trikutnik ACH similar to trikutnik AB Three kutami. similarly tricot CBH similar ABC.

Introduced designations:

we accept:

what do you think -

sklavshi a 2 that b 2, we accept:

otherwise, what was needed to bring.

2. Confirmation of the Pythagorean theorem by a path area.

Prove lower, no matter how simple it is, but it’s not so simple. All stinks

win the power of the area, prove some folding for the proof of the Pythagorean theorem itself.

Proof through equivalence.

Roztashuemo chotiri equal straight

trikutnik so, as shown in the little one

right-handed

Chotirikutnik with sides c- square,

oskіlki sum of two gostrih kutіv 90 °, and

flared cut - 180 °.

The area of all figures is healthy, from one side,

area of the square with its side ( a+b), and from the other side, the sum of the area of chotiriokh trikutnikov i

What did it take to bring.

3. Proof of the Pythagorean theorem by the infinitely small method.

Looking at the armchair, showing at the little one, and

guarding the change of sidesa, we can

write down the next date for indefinitely

malich side incrementhі a(vicory similar to

trikutnikov):

Vikoristovuyuchi method podіlu zminnyh, we know:

Greater global virase change hypotension in different increments of both catheters:

Integrating the data, the victorious cob, mind, we will:

In this order, we come to the Bazhan vіdpovіdі:

It doesn’t matter how much, the quadratic fallacy of the residual formula is linear

proportionality between the sides of the tricot and increments, even though the sum is tied with independent

contributions in the form of zbіlshennya rіznih catetіv.

The simplest proof can be taken away, as to consider that one of the catheti is not considered greater

(in this vipadka leg b). The same for the constant integration is taken:

According to Van der Waerden's thought, it is even more remarkable that spіvvіdnoshennia in a slanderous way was in Babylon's house already close to the 18th century BC. e.

Approximately 400 roku BC. That is, according to Proclus, Plato gave a method for the recognition of Pythagorean triplets, which is based on algebra and geometry. Close to 300 rock to star. That is, in the "Cobs" of Euclid, the oldest axiomatic proof of the Pythagorean theorem appeared.

Formula

The main formulation to revenge the algebraic diї - for a straight-cut tricot, dozhini catheti of any equal a (\displaystyle a)і b (\displaystyle b), and the length of the hypotension - c (\displaystyle c), Vikonano spіvvіdnoshennia:

Possible and equivalent geometric formulary, which goes into understanding the area of \u200b\u200bfigures: in a rectangular tricot, the area of \u200b\u200bthe square, pobudovannaya on the hypotenuse, more sum of the area of the squares, pobudovanih on the legs. In this view, the theorem is formulated on Euclid's Ears.

Pythagoras' turnaround theorem- a statement about the uprightness of any kind of knitter, the life of the side of any kind of pov'yazan a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)). Like a last, for a triple of positive numbers a (\displaystyle a), b (\displaystyle b)і c (\displaystyle c), such that a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2))іsnuє straight cut tricout with legs a (\displaystyle a)і b (\displaystyle b) that hypotenuse c (\displaystyle c).

Prove

The scientific literature has recorded less than 400 proofs of the Pythagorean theorem, which, like fundamental values for geometry, is elementary in the result. The main direct proofs are: the variant of algebra spіvvіdnoshen elements-trikutnik (such, for example, the popular method of similarity), the method of area, and also different exotic proofs (for example, for the help of differential rivnіan).

Through similar tricks

Euclid's classic proof was straightened by the establishment of evenness of the area between the rectangles, set up with a square gap over the hypotenuse height from a straight kut with squares over the legs.

Design, as a victorious for proof: for a straight-cut tricot with a straight cut C (\displaystyle C), square over the legs and square over the hypotenuse A B I K (\displaystyle ABIK) height will be C H (\displaystyle CH) and promin, what її continue, s (\displaystyle s), which divides the square above the hypotenuse into two rectangles i . Proof of orientation on the installation of evenness of the area of the rectangle A H J K (\displaystyle AHJK) with a square over the leg A C (\displaystyle AC); the equality of the area of another rectangle, which makes a square over the hypotenuse, that rectangle above the other leg is installed in a similar rank.

Rivnist area of a rectangle A H J K (\displaystyle AHJK)і A C E D (\displaystyle ACED) be restored through the congruence of tricots △ A C K (\displaystyle \triangle ACK)і △ A B D (\displaystyle \triangle ABD) area of the skin with some half of the area of the squares. A H J K (\displaystyle AHJK)і A C E D (\displaystyle ACED) apparently at the link with the advancing power: the area of the tricot is the half of the square of the rectangle, so the figure is the double side, and the height of the tricot to the upper side is the other side of the rectangle. The congruence of knitwear is evident from the evenness of two sides (sides of squares) and a cut between them (folded from a straight cut and a cut at A (\displaystyle A).

In such a rank, the proof is established that the area of the square above the hypotenuse, which is made up of rectangles A H J K (\displaystyle AHJK)і B H J I (\displaystyle BHJI), what is the sum of squares above the legs.

Proof of Leonardo da Vinci

Before the method of the square, there is also proof of the knowledge of Leonardo da Vinci. Let it be given a straight-cut tricoutnik △ A B C (\displaystyle \triangle ABC) with a straight cut C (\displaystyle C) that square A C E D (\displaystyle ACED), B C F G (\displaystyle BCFG)і A B H J (\displaystyle ABHJ)(Div. babies). Who has evidence on the side H J (\displaystyle HJ) the rest of the old bik will have a trikutnik, congruent △ A B C (\displaystyle \triangle ABC), before that, it was like a hypotenus, and how high to it (tobto J I = B C (\displaystyle JI = B C)і H I = A C (\displaystyle HI=AC)). Straight C I (\displaystyle CI) breaks the square, promptings on the hypotenuse on two equal parts, shards of trikutnik △ A B C (\displaystyle \triangle ABC)і △ J H I (\displaystyle \triangle JHI) wake up. The proof establishes the congruence of chotirikutniks C A J I (\displaystyle CAJI)і D A B G (\displaystyle DABG), the area of the skin z yak, appear, from one side, more than the sum of half the area of the squares on the legs and the area of the lateral tricot, from the other side - half the area of the square on the hypotenuse plus the area of the extrinsic tricot. Also, half the sum of the areas of the squares above the legs is equal to the half of the area of the square above the hypotenuse, which is the same as the geometric formulation of the Pythagorean theorem.

Proof by infinitely small method

I have a piece of evidence that goes into the technique of differential equations. Zokrema, Hardy is credited with a proof that the victorist’s infinitely small increase in catheters. a (\displaystyle a)і b (\displaystyle b) and hypotension c (\displaystyle c) and take care of the similarity with the outward straight line, so that it will prevent the coming differential speeds:

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

By the method of sub-dimensions of їх, differential equalization is deduced c d c = a d a + b d b (displaystyle c dc = a, da + b, db), integration c 2 = a 2 + b 2 + C o ns t (\displaystyle c^(2)=a^(2)+b^(2)+\mathrm (Const) ). Zastosuvannya cob minds a = b = c = 0 (\displaystyle a=b=c=0) we assign a constant like 0, which results in the confirmation of the theorem.

Quadratic deposits in the residual formula are the lines of linear proportion between the sides of the tricot and the increments, even though the amount is due to the independent contributions from the increments of the other categories.

Variations and zagalnennya

Similar geometric figures on three sides

An important geometrical elaboration of the Pythagorean theorem was given by Euclid in "The Cobs", passing from the areas of squares on the sides to the areas of additional similar geometric figures: ї on the hypotenuse.

The main idea of \u200b\u200bthe whole idea is based on the fact that the area of \u200b\u200blike a geometric figure is proportional to the square of its own linear size and square to the square of its own side. Otzhe, for similar figures from Maidans A (\displaystyle A), B (\displaystyle B)і C (\displaystyle C), pobudovanih on legs іz dozhina a (\displaystyle a)і b (\displaystyle b) and hypotenuse c (\displaystyle c) vodpovidno, may be spivvidnoshnya:

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

Bo behind the Pythagorean theorem a 2 + b 2 = c 2 (\displaystyle a^(2)+b^(2)=c^(2)), then viconano.

In addition, it is possible to bring the Pythagorean theorem without abbreviation, that for the areas of three similar geometric figures on the sides of a rectangular tricutnik A + B = C (\displaystyle A+B=C), then from the reverse course of the confirmation of Euclid's proof, one can prove the proof of the Pythagorean theorem. For example, as if on the hypotenuse, induce a congruent cob straight-cut tricot with a square C (\displaystyle C), and on the legs - two straight-cut tricots similar to it with squares A (\displaystyle A)і B (\displaystyle B), then it appears that the tricots on the legs are settled as a result of the subdivision of the cob trikutnik with its height, that is the sum of two smaller areas of the trikutniks in the larger area of the third, in such a rank A + B = C (\displaystyle A+B=C) and, zastosovuyuchi spіvvіdnoshennia for such figures, the Pіthagorean theorem is introduced.

Cosine theorem

The Pythagorean theorem is a perfect example of a greater than the general theorem of cosines, as it proves to be true of the sides of a good trickster:

a 2 + b 2 − 2 a b cos ⁡ θ = c 2,

de - kut between parties a (\displaystyle a)і b (\displaystyle b). Yakshcho kut dorivnyuє 90 °, then cos ⁡ θ = 0 (\displaystyle \cos \theta =0) and the formula is reduced to the ultimate Pythagorean theorem.

Dovіlny tricoutnik

It is important to note that Pythagoras's theorem is based on a more tricutnik, which operates exclusively on the spіvvіdshennym dovzhin storіn, it is important that it was first installed by the Sabiysk astronomer Sabit Ibn Kurroy. In the new one for a long tricot with sides to the new one, an equal-femoral tricot with a base on the side c (\displaystyle c), the top, which runs from the top of the outward tricutnik, which lies along the sides c (\displaystyle c) and kutami with substantiation, equal kutu θ (\displaystyle \theta), protileous boci c (\displaystyle c). At the result, two tricots are established, similar to the last one: the first one - on the three sides a (\displaystyle a), far away from the side of the inscribed equal-femoral tricot, that r (\displaystyle r)- part of the side c (\displaystyle c); the other is symmetrical to the new side b (\displaystyle b) on the side s (\displaystyle s)- double side c (\displaystyle c). As a result, there is a vikonan spіvvіdnosheniya:

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

which translates into the Pythagorean theorem when θ = π / 2 (\displaystyle \theta =\pi /2). Spivvіdnoshnja є naslіbnosti utavnіh trikutnikov:

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

Pappus' area theorem

Non-Euclidean geometry

The Pythagorean theorem is based on the axes of Euclidean geometry and is not valid for non-Euclidean geometry - the Pythagorean theorem is more equal to the postulate of Euclidean parallelism.

In non-Euclidean geometry, the alignment between the sides of a rectangular tricot will be obov'yazkovo in the form, as it is in agreement with the Pythagorean theorem. For example, in a spherical geometry, all three sides of a rectangular tricot, as if they are surrounded by an octant of a single sphere, can π / 2 (\displaystyle \pi /2), what is the superstition of the Pythagorean theorem

For this reason, the Pythagorean theorem is valid in hyperbolic and elliptic geometry, so that I can replace the straightness of the tricot with the mind one, because the sum of the two kutіv of the tricot is to be added to the third.

Spherical geometry

For any rectangular tricot on a sphere with a radius R (\displaystyle R)(for example, yakscho kut at trikutnik straight) with 3 sides a, b, c (\displaystyle a, b, c) spіvvіdnoshennia between the parties may look:

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

This equality can be shown as a special feature of the spherical cosine theorem, as it is valid for all spherical tricots:

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

de ch (\displaystyle \operatorname (ch) )- Hyperbolic cosine. Tsya formula є let's call it a type of hyperbolic cosine theorem, as it is valid for all tricks:

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

de γ (\displaystyle\gamma)- Kut, the top of which is on the opposite side c (\displaystyle c).

The Taylor series for the hyperbolic cosine ( ch ⁡ x ≈ 1 + x 2 / 2 (\displaystyle \operatorname (ch) x\approx 1+x^(2)/2)) you can show how the hyperbolic tricoutnik changes (if a (\displaystyle a), b (\displaystyle b)і c (\displaystyle c) to zero), then the hyperbolic extensions of a straight-cut tricutnik are approaching the extensions of the classical Pythagorean theorem.

Zastosuvannya

Vіdstanov in dvuhіrіnіh pryamokutnyh systems

The most important application of the Pythagorean theorem is the determination of the difference between two points in a rectangular, system, coordinates: s (\displaystyle s) between points with coordinates (a,b) (\displaystyle (a,b))і (c, d) (\displaystyle (c, d)) one:

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

For complex numbers, the Pythagorean theorem gives a natural formula for the value of the modulus of a complex number - for z = x + y i (\displaystyle z = x + yi) wine

Theorem

In a straight-cut tricot, the square of the length of the hypotenuse is more than the sum of the squares of the length of the legs (Fig. 1):

$c^(2)=a^(2)+b^(2)$

Proof of the Pythagorean theorem

Let the tricout $A B C$ be a straight tricout with a straight tie $C$ (Fig. 2).

Let's draw a height from the top $ C $ to the hypotenuse $ A B $, the base of the height is meaningfully $ H $.

Rectangular tricout $A C H$ is similar to tricut $A B C$ in two folds ($\angle A C B=\angle C H A=90^(\circ)$, $\angle A$ is a rounded one). Similarly, $C B H$ is similar to $A B C$ .

Vivshi sign

$$B C=a, A C=b, A B=c$$

s similar trikutnikov otrimuemo, scho

$$\frac(a)(c)=\frac(H B)(a), \frac(b)(c)=\frac(A H)(b)$$

Zvіdsi maєmo, scho

$$a^(2)=c \cdot H B, b^(2)=c \cdot A H$$

Squeeze away zealousness, take away

$$a^(2)+b^(2)=c \cdot H B+c \cdot A H$$

$$a^(2)+b^(2)=c \cdot(H B+A H)$$

$$a^(2)+b^(2)=c \cdot A B$$

$$a^(2)+b^(2)=c \cdot c$$

$$a^(2)+b^(2)=c^(2)$$

What did it take to bring.

Geometrical formulation of the Pythagorean theorem

Theorem

For a straight-cut tricot of the area of the square, induced on the hypotenuse, more sum of the areas of the squares, induced on the legs (Fig. 2):

Apply the solution of tasks

butt

Manager. The task is a rectilinear tricot $A B C$, the legs of which are 6 cm and 8 cm long. Find the hypotenuse of this tricot.

Solution. Zgіdno z umovoy leg $a=6$ cm, $b=8$ cm.

$c^(2)=a^(2)+b^(2)=6^(2)+8^(2)=36+64=100$

It must be taken into account that the hypotenuse is stupid

$c = \sqrt(100) = 10$(cm)

Vidpovid. 10 cm

butt

Manager. Find the area of a straight-cut tricot, as it is clear that one of the legs is 5 cm larger than the other, and the hypotenuse is 25 cm longer.

Solution. Let $x$ cm - the length of the smaller leg, then $(x+5)$ cm - the length of the larger one. The same thing with the Pythagorean theorem can be:

$$x^(2)+(x+5)^(2)=25^(2)$$

Opening the arches, making similar ones and virishuemo otrimane square alignment:

$x^(2)+5 x-300=0$

Vіdpovіdno up to the theorem Вієт, otrimuєmo, scho

$x_(1)=15$ (cm) , $x_(2)=-20$ (cm)

The value of $x_(2)$ does not satisfy the minds of the mind, therefore, the smaller leg is 15 divs, and the larger one is 20 divs.

The square of a straight-cut tricot fabric is more like a dozhin yoga cathetiv, tobto

$$S=\frac(15 \cdot 20)(2)=15 \cdot 10=150\left(\mathrm(cm)^(2)\right)$$

Vidpovid.$S=150\left(\mathrm(cm)^(2)\right)$

Historical proof

Pythagorean theorem- One of the main theorems of Euclidean geometry, which establishes a parallel between the sides of a rectangular tricot.

The ancient Chinese book "Zhou bi suan jing" has a Pythagorean tricot with sides 3, 4 and 5. The greatest German historian of mathematics Moritz Kantor (1829 - 1920) knows that $3^(2)+4^(2)=5 ^ (2) $ was already known to the Egyptians as early as 2300 BC. As far as I thought, alarm clocks were still straight along for the help of straight-cut tricots with sides 3, 4 and 5. More about the Pythagorean theorem among the Babylonians. In one text, a calculation of the hypotenuse of a rіvnofemoral straight-cut tricutnik is brought closer.

At the moment, 367 proofs of the theorem have been recorded in the scientific literature. Imovirno, the Pythagorean theorem is a single theorem with a significant number of proofs. Such a difference can be explained only by the fundamental meanings of the theorem for geometry.

The potential for creativity should be attributed to the humanities disciplines, naturally scientifically rich in analysis, practical ideas and dry language of formulas and figures. Mathematics is not up to humanities subjects. Ale without creativity in the "queen of all sciences" will not go far - people know about it a long time ago. Three hours of Pythagoras, for example.

Shkіlnі podruchniki, unfortunately, don't call out to explain that in mathematics it is important not only to cram theorems, axioms and formulas. It is important to understand and understand the fundamental principles. And when you try to learn your mind from stamps and abetic truths - it’s only in such minds that all great ideas are people.

To such conclusions, one can add those who today know the Pythagorean theorem. With this help, we will try to show that mathematics is not only able, but I can be a fool. And this suit is suitable not only for botanists at such eyepieces, but for everyone who has a strong mind and a strong spirit.

From the history of nutrition

Strictly ostensibly, even though the theorem is called "the Pythagorean theorem", Pythagoras himself did not prove it. The straight-cut tricoutnik and the special powers wriggled back to the new one. Take a look at the food chain. For one version, Pythagoras was the first to know the complete proof of the theorem. For any other proof, the authorship of Pythagoras cannot be attributed.

Today, you won’t overthink who has a walk and who has mercy. Vіdomo less, scho prove Pіthagoras, yakscho won, if it fell asleep, it was not saved. Vtіm, there are excuses that the famous proof from the “Pochatkіv” of Euclid may belong to Pіthagoras, and Euclid yogo only fixed.

It is also seen today that the story about the straight-cut knitwear is written in the Egyptian clocks of Pharaoh Amenemhet I, on the Babylonian clay tablets during the reign of King Hammurabi, in the ancient Indian treatise "Sulva Sutra" that ancient Chinese to whom you create "Zhoubi-sun".

Like Bachite, the Pythagorean theorem has occupied the minds of mathematicians for the most recent hours. To confirm, there are close to 367 different proofs, which are confirmed today. For whom it cannot compete with another theorem. Among the famous authors of evidence, one can guess Leonardo da Vinci and the twentieth US President James Garfield. It's all worth talking about the supra-primary importance of the theorem for mathematics: it should be shown, otherwise, more theorems of geometry are connected with it.

Prove the Pythagorean theorem

For school assistants, it is important to induce algebraic proofs. But the essence of the theorem in geometry, then let's look at us in front of you and prove the famous theorem, which is based on this science.

Proof 1

For the simplest proof of the Pythagorean theorem for a rectangular tricot, it is necessary to establish an ideal mind: let the tricot be not only rectilinear, but also equal-femoral. Let me remind you that such a trickster himself looked at the mathematics of old.

Confirmation "square, promptings on the hypotenuse of a rectangular tricot, an equal sum of squares, promptings on yoga legs" can be illustrated by the upcoming armchairs:

Marvel at the even-femoral straight-cut tricot ABC: On the hypotenuse AC, you can induce a square, which is made up of several tricots, which is good for the outer ABC. And on the legs AB and PS it was suggested in a square, the skins of these should be replaced by two similar tricots.

Before the speech, the christening formed the basis of numerous anecdotes and caricatures, attributions to the Pythagorean theorem. The most famous, maybe, tse "Pythagorean pants on all sides":

Proof 2

This method is combined with algebra and geometry and can be considered as a variant of the ancient Indian proof of the mathematician Bhaskari.

Stay a straight-cut tricout with 3 sides a, b and c(Fig.1). Let's remember two squares with sides, equal to the sum of the dozhins of two catheti, - (a+b). At the skin of the squares, wake up, like in figures 2 and 3.

At the first square, make chotir of such tricks, like on a little one 1. The result will have two squares: one with side a, the other with side b.

In another square, chotiri prompted similar tricots to make a square on the other side, which is the most favorable hypotenuse. c.

The sum of the squares of the squares in Fig. 2 is equal to the area of the square we have worked out with side z in Fig. 3. It's easy to overthink it by changing the area of the squares in fig. 2 behind the formula. And the area of the inscribed square on the little one 3. by the way, the area of \u200b\u200bchotiriox equal between themselves is inscribed at the square of rectangular tricots from the area of \u200b\u200bthe great square on the other side (a+b).

Having written down everything, maybe: a 2 + b 2 \u003d (a + b) 2 - 2ab. Open the arms, carry out the necessary algebraic calculations and take away what a 2 + b 2 = a 2 + b 2. At tsiom the area inscribed in Fig.3. squares can be calculated using the traditional formula S=c2. Tobto. a2+b2=c2- You have completed the Pythagorean theorem.

Proof 3

The very old Indian proof of the descriptions in the XII century in the treatise “Vinets znannya” (“Siddhanta shiromani”) and as the main argument the author of the victor’s call, turning to mathematical talents and watchfulness of the scholars and followers: “Wow!”.

Ale mi, let's take a look at the proof of the greater report:

In the middle of the square, stay chotiri straight-cut tricutniks as it is indicated on the armchair. The side of the great square, there is a hypotenuse, significantly h. Trikutnik's legs are called Aі b. Vіdpovіdno to the armchair side of the inner square (a-b).

Find the formula for the area of a square S=c2, to calculate the area of the outer square. І one hour, change the same value by adding the area of the inner square and the area of all four rectangular tricots: (a-b) 2 2+4*1\2*a*b.

You can choose between different options for calculating the square area, so that you can reconsider so that they give the same result. I give you the right to write down what c 2 =(a-b) 2 +4*1\2*a*b. As a result of the decision, you take the formula of the Pythagorean theorem c2=a2+b2. The theorem has been completed.

Proof 4

This ancient Chinese proof, having omitted the name “The Chosen Stelete” - through I go to the stele figure, as a result of all promptings:

The new one wins chairs, as we have already succumbed in Fig. 3 to another proof. And the inner square with the side of the promptings is the same, as in an old Indian proof, induced more.

As an idea to fit the armchair in Fig. 1, two green straight-cut tricots, move them to the opposite sides of the square with the side z and hypotenuses, apply to the hypotenuses of the buzkovy trikutniks, and then the figure under the name "steletz named" (Fig. 2). For accuracy, you can do the same with paper squares and tricots. You are confused that the "betrothed man" is made up of two squares: small ones on the side b i great zі side a.

Qi pobudovi allowed the old Chinese mathematicians and we were to follow them c2=a2+b2.

Proof 5

Another way to know the solution for the Pythagorean theorem is to rely on geometry. It is called "Garfield Method".

Stay a straight-cut tricoutnik ABC. We need to bring what ND 2 \u003d AC 2 + AB 2.

For whom to continue the leg AC and stay awake CD, which is a better leg AB. Drop Perpendiculars AD vіdrіzok ED. Vіdrіzki EDі AC equal. Connect points Eі At, as well as Eі W and take away the armchair, like a little lower:

In order to bring the tower, we again go to the method we have already tried: we know the area of \u200b\u200bthe figure that we have seen, in two ways and compare it one to one.

Know the square of the bagatokutnik ABED it is possible, having folded the area of three trikutniks, like її utvoryuyut. And one of them ECB, not only upright, but rіvnofemoral. Let's not forget that AB = CD, AC = EDі BC = PЄ- do not let us ask you to write down not to overwhelm Yogo. Otzhe, S ABED \u003d 2 * 1/2 (AB * AC) + 1 / 2BC 2.

With whom, it is obvious that ABED- Trapezium. Therefore, I calculate the area for the formula: S ABED = (DE + AB) * 1/2AD. For our calculation, it’s easier and better to show the winds AD like a sum of money ACі CD.

Let's write down the offensive ways to calculate the area of the figures, putting a sign of equality between them: AB*AC+1/2BC 2 =(DE+AB)*1/2(AC+CD). Vikoristovuєmo already known to us and described more equivalence of vіdrіzkіv, to forgive the right part of the record: AB*AC+1/2BC 2 =1/2(AB+AC) 2. And now we open the arches and transform the equanimity: AB*AC+1/2BC 2 =1/2AC 2 +2*1/2(AB*AC)+1/2AB 2. Having completed all the transformations, we take away those that we need: ND 2 \u003d AC 2 + AB 2. We brought the theorem.

Of course, this list of evidence is far from complete. The Pythagorean theorem can also be brought to the aid of vectors, complex numbers, differential equalities, stereometry too. I teach physics: as, for example, in analogous representations on square chairs and tricot obsyagi, fill the motherland. By pouring the motherland, one can bring the equality of the area and the very theorem of the result.

A couple of words about the Pythagorean triplets

There are few people who don’t get caught up in the school program. And this time of the day, the arch of the cicavia and may is of great importance in geometry. Pythagorean triplets zastosovuyutsya for the accomplishment of rich mathematical problems. The announcement about them may be good for you in the future.

So what are the Pitagorian triplets? This is the name of the natural numbers, chosen by three, the sum of the squares of two of them is equal to the third number in the square.

Pythagorean triplets can be:

primitive (all three numbers are mutually simple);
not primitive (like the skin number of the trio multiplied by the same number, we will see a new triple, like not a primitive one).

Even before our time, the ancient Egyptians were fascinated by the mania of the numbers of Pythagorean triplets: at the stinks of the stench, they could see a straight-cut tricot with sides of 3.4 and 5 ones. Before the speech, be it a trickster, the sides of which are equal to the numbers from the Pitagorean trinity, for the locks are straight-cut.

Apply the Pythagorean trios: (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 proof of the theorem

The Pythagorean theorem is known to be stuck in mathematics, and in architecture and everyday life, astronomy and inspiring literature.

A note about everyday life: to know the Pythagorean theorem at a new wide station at the heads of a different level of folding. For example, marvel at the Romanesque style:

Significantly the width of the window is b the same radius of the great pіvkol can be recognized as R and through b: R=b/2. The radius of smaller pіvkol is also visible through b: r=b/4. At tsomu zavdannya we have to cackle the radius of the inner stake of the vekna (called yogo p).

Pythagorean theorem R. For whom the vikorist is a straight-cut tricouter, but for a little bit of meanings a dotted line. The hypotenuse of the tricot is composed of two radii: b/4+p. One leg is a radius. b/4, іnsh b/2-p. Vikoristovuyuchi Pіthagorean theorem, we write: (b/4+p) 2 =(b/4) 2 +(b/2-p) 2. Dali rozkriёmo bows and take away b 2 /16+ bp / 2 + p 2 \u003d b 2 / 16 + b 2 / 4-bp + p 2. Let's turn this viraz on bp/2=b 2 /4-bp. And then we will divide all the members into b, we will suggest similar, we will take away 3/2*p=b/4. As a result, we know that p=b/6- What do we need.

For an additional theorem, you can calculate a double krokvi for a double dahu. Significantly, the height of the mobile phone connection is necessary, so that the signal reaches the song settlement. And to build a new yalinka steadily on the Moscow Maidan. Like a bachite, this theorem lives not only on the sides of assistants, but often becomes a corollary to real life.

As far as literature is concerned, the Pythagorean theorem suffocated the writers for hours of antiquity and continues to work for our time. For example, the German writer of the 19th century, Adelbert von Chamisso, sighed at the writing of the sonnet:

The light of truth will not rise soon,
Ale, having declared, it is unlikely to grow up
I, like a thousand years ago,
Do not call sumnіvіv i superechki.

Naimudrishі, if I look
The light of truth, the gods speak;
I hundred beats, stabbed, lie down.
A gift from the confession of Pythagoras.

From that hour on, the beats roared wildly:
Naviki alarmed the bichache tribe
Podiya, guessed here.

Let us give up: axis-axis, the hour has come,
I want to make a sacrifice again
What a great theorem.

(Translated by Viktor Toporov)

And in the 20th century, the Radianian writer Evgen Veltistov at the book “Fit Electronics” proved the Pythagorean theorem, proving the Pythagorean theorem. And more and more rozpovіdі about the world of two worlds, what a moment it was to realize, like the Pythagorean theorem became the basic law and to instill the religion of the whole world taken. Life in the new world would be a lot easier, ale and a lot more boring: for example, there is no understanding of the meaning of the words “round” and “fluffy”.

And also in the book "Fit Elektronika" the author, with the words of the teacher of mathematics Taratar, seems to say: "Mathematics has a head - thoughts, new ideas." The very creative policy of the mind gives birth to the Pythagorean theorem - it is not for nothing that it has many different proofs. Vaughn helps to look beyond the vowel and marvel at the familiar speech in a new way.

Visnovok

This article has been created so that you can look at the inter-school programs of mathematics and find out more about proving the Pythagorean theorem, as instructed in the handbooks "Geometry 7-9" (L.S. Atanasyan, V.M. Rudenko) and "Geometry 7 -11 ”(A.V. Pogorelov), ale and іnshі tsіkavі ways to bring the famous theorem. And also apply it, like the Pythagorean theorem can zastosovuvatsya in an extraordinary life.

First of all, this information will allow you to apply for the highest score in mathematics lessons - the results of the subject from the supplementary necks are always highly valued.

In a different way, we wanted to help you to see how much mathematics is a science. Perekonatisya on specific butts, scho zavzhd є m_sce creativity. We suspect that the Pythagorean theorem is such an article to inspire you on independent jokes and bad reviews in mathematics and other sciences.

Tell us in the comments, what did you find in the article, prove it. You needed qi at the head office. Write to us what you think about the Pythagorean theorem and this article - we will be happy to discuss everything with you.

blog.website, with a new or private copy of the material sent on the original binding.

Pythagorean theorem: Sum of square squares that spiral on the legs ( aі b), the additional area of the square, called on the hypotenuse ( c).

Geometric formulation:

Back on the cob, Boule's theorem is formulated in an offensive rank:

Algebraic formula:

Tobto, knowing the length of hypotension of the tricutnik through c, and dozhini cathetiv through aі b :

a 2 + b 2 = c 2

Offensive formulation of the theorem is equivalent, but otherwise the formulation is more elementary, it does not mean understanding the area. To that another firmness can be distorted, not knowing anything about the area, having only died down the sides of a straight-cut tricutnik.

Pythagoras' turnaround theorem:

Prove

Understanding, conceptually they can be divided into a small number of classes. Among them: prove by the area method, axiomatic and exotic proofs (for example, for the help of differential equations).

Through similar tricks

The forthcoming proof of the algebraic formula is the simplest of the proofs, which are directly behind the axioms. Zokrema, it’s not victorious to understand the figure’s area.

Come on ABCє straight cut tricout with a straight cut C. Let's draw a height from Cі meaningfully її basis through H. Trikutnik ACH similar to trikutnik ABC in two sections. similarly tricot CBH similar ABC. Vivshi sign

acceptable

What is equivalent

Squeal, take it

Prove by area method

Prove lower, no matter how simple it is, but it’s not so simple. All stinks win the power of the area, prove some folding for proving the Pythagorean theorem itself.

Proof through equivalence

Roztashuemo chotiri straight-cut tricutniks so, as shown in the little one 1.
Chotirikutnik with sides cє square, the shards of the sum of two gostrikh kutiv 90 °, and the open kut - 180 °.
The area of all the figures is solid, from one side, the area of the square with the side (a + b), from the other side, the sum of the squares of the tricots and two inner squares.

What did it take to bring.

Prove through equivalence

An elegant proof behind an additional rearrangement

The butt of one of these proofs is indicated on the right-handed armchair, de square, prompting on the hypotenuse, permuting into two squares, prompting on the legs.

Euclid's proof

Armchair before the proof of Euclid

Illustration before Euclid's proof

The idea of proving Euclid is in the offensive: we will try to bring that half of the area of the square, pobudovannaya on the hypotenuse, more than the sum of the halves of the area of the squares, pobudovannyh on the legs, and even the area of the great and two small squares equal.

Let's take a look at the armchair. On the new mi, we made squares on the sides of a rectangular tricorne and drew from the top of the straight coil C promin perpendicularly to the hypotenuse AB, rose square ABIK, prompted on the hypotenuse, on two rectangles - BHJI and HAKJ obviously. It appears that the areas of these rectangles are exactly equal to the areas of the squares that were placed on the right legs.

Let's try to bring out that the area of the square DECA is equal to the area of the rectangle AHJK The last thing is the designation of the area of the tricot as half of the foundation of the height. From the first warning you see that the area of the tricot ACK is more than the area of the tricot AHK (not depicted on the little one), yak, near its back, the half of the area of the straight cut AHJK.

Now let's say that the area of the tricot ACK is also half the area of the DECA square. The only thing that is necessary for this purpose is to bring the equality of the tricots ACK and BDA (the shards of the area of the tricot BDA are equal to half the area of the square for the ordered power). Equality is obvious, tricots are equal on both sides and a kutku between them. Same - AB=AK,AD=AC - the evenness of the cuts CAK and BAD is easy to bring by the method of movement: turn the CAK tricout 90 ° opposite the arrow, it’s also obvious that the sides of the two tricouts, which are looked at, widen (through the kut at the top of the square - 90°).

The observation about the evenness of the area of the square BCFG and the rectangle BHJI is absolutely similar.

Tim himself brought it to light that the area of a square, induced on the hypotenuse, is formed from the area of the squares, induced on the legs. The idea of \u200b\u200bwhich proof is additionally illustrated for additional animation, as it is more elaborate.

Proof of Leonardo da Vinci

The main element of proof is symmetry and ruh.

Look at the armchair, as you can see from symmetry, vіdrіzok CI pink square ABHJ on two same parts (shards of tricutniks ABCі JHI equal for the pobudova). Koristuyuchisya turn by 90 degrees against the year's arrow, we vbachayemo evenness of the hatched figures CAJI і GDAB . Now it is clear that the area of the figure shaded by us is more than the sum of the halves of the area of \u200b\u200bthe squares that were on the legs, that the area of \u200b\u200bthe outward tricot. On the other side, on the other half of the area of the square, which was placed on the hypotenuse, plus the area of the outer tricot. The rest of the proof is hoped to be read.

Proof by infinitely small method

The offensive proof for the help of differential equations is often attributed to the famous English mathematician Hardy, who is alive in the first half of the 20th century.

Looking at the armchair, showing at the little one, and watching the change of side a, we can record the onset of spіvvіdnoshenya for infinitely small increments of sides hі a(vicorist similar to trikutnikov):

Proof by infinitely small method

Koristuyuchis by the method of podіlu zminnyh, we know

More gal virase change hypotension in different increments of both legs

Integrating the given equal and victorious cobs, mind, otrimuemo

c 2 = a 2 + b 2+ constant.

In this order, we come to the Bazhan vіdpovіdі

c 2 = a 2 + b 2 .

No matter how bigity, the quadratic deposit in the residual formula is the lines of linear proportion between the sides of the tricot and the increments, even though the sum is due to the independent deposits in the increment of the other categories.

The simplest proof can be taken as evidence that one of the legs is not growing (in this case, the legs b). Then for the constant integration is taken

Variations and zagalnennya

If you want to replace the squares on the legs of other similar figures, then the Pythagorean theorem will come true: A straight-cut tricot has a sum of areas of similar figures, induced on the legs, more areas of figures, induced on the hypotenuse. Zokrema:
- The sum of the area of regular tricots, awakened on the legs, the additional area of the correct trikutnik, awakened on the hypotenuse.
- The sum of the area of the pivkolіv, pobudovannyh on the legs (like the diameter), the dorіvnyuє pіvkolіv, pіvkolіv, pobudovanogo on the hypotenuse. This butt is victorious when proving the power of the figures, surrounded by arcs of two kilos and wearing the name of hippocratic moons.

History

Chu-Pei 500-200 BC Zliva wrote: the sum of the squares of the length of the height and the base is the square of the length of the hypotension.

In the ancient Chinese book Chu-Pei, there is a story about Pythagorean tricot with sides 3, 4 and 5: In this book, there is a proponation of little ones, which runs with one of the armchairs of the Indian geometry of Bashary.

Kantor (the greatest German historian of mathematics) knows that the equality of 3 + 4 + 5 = was already known to the Egyptians as early as 2300 BC. e .., for the hours of King Amenemhat I (gidno with papyrus 6619 to the Berlin Museum). According to Kantor's idea, harpedonapti, or pulling bobbins, were straight kuti for the help of straight-cut tricutniks with sides 3, 4 and 5.

It’s even easier to follow the same way. Let's take a windbreaker 12 m long and tie it to her according to her husband's color on a 3 m long. from one quarter and 4 meters from another. A straight kut will appear to be connected between the sides of the zavdovka 3 and 4 meters. The harpedonapts could be rebuffed, that their way would encourage us to become occupied, as if to speed up, for example, with a wooden mower, which would be stagnant with all the joiners. There are Egyptian little ones, on which such a tool is used, for example, little ones that represent a joiner's work.

There is more information about the Pythagorean theorem among the Babylonians. In one text, which is dated until the hour of Hammurabi, that is until 2000 BC. That is, to be guided closer to the calculation of the hypotenuse of a straight-cut tricutnik. Zvіdsi you can make a nevtіshny visnovka, that at the Dvorichchya they were able to work with straight-cut tricutniks, at the extreme approach of some vipadkas. Grounded, on the one hand, on the current knowledge of Egyptian and Babylonian mathematics, and on the other hand, on the critical wreathing of walnuts, van der Waerden (Dutch mathematician) made such a winding:

Literature

Russian mine

Skopets Z. A. Geometric miniatures. M., 1990
Yelensky Sh. Following Pythagoras. M., 1961
Van der Waerden B. L. Awakened science. Mathematics of Ancient Egypt, Babylon and Greece. M., 1959
Glazer G.I. History of mathematics at school. M., 1982
St Litzman, Pythagorean Theorem, Moscow, 1960.
- A site about the Pythagorean theorem with a great number of proofs of the material taken from the book of V. Litzman, a large number of armchairs are presented in a large number of graphic files.
The Pythagorean theorem and the Pythagorean triplets, chapter 3 of the book by D. V. Anosov “A look at mathematics and see it”
About the Pythagorean theorem and ways of її proof G. Glazer, Academician of the Russian Academy of Arts, Moscow

English

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

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