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Everything you need to know about trikutnik. The first sign of jealousy of the trikutniks. Second and third signs of fidelity of the tricutaneous Trigonometric functions of the external cut

Among the large number of rich cutlets, which are essentially a closed laman line that does not fray, the tricutnik is the figure with the smallest number of cutlets. In other words, this is the simplest rich man. However, despite all its simplicity, it still conceals a lot of mysteries and secrets that are illuminated by a special branch of mathematics - geometry. This discipline in schools begins to be extended from this class, and the topic “Trikutnik” is given special respect here. Children not only learn the rules about the figure itself, but also follow them, including 1, 2 and 3 signs of jealousy.

First knowledge

One of the first rules that schoolchildren are familiar with sounds approximately like this: the sum of the sizes of all three cutlets is equal to 180 degrees. To confirm this, it is enough to use a protractor to measure the skin from the tops and folds of all the values ​​that came out. Coming from this, it is easy to calculate the third for two given quantities. For example: For a trikutnik, one of the kuts is 70 °, and the other is 85 °, what is the size of the third kut?

180 - 85 - 70 = 25.

Version: 25°.

Orders can be even more complex, since only one value is indicated, and less is said about another value, how many or how many times there is more or less.

To identify these and other features of the tricube, special lines can be drawn, the skin of which gets its name:

  • height – perpendicular to the straight line, drawn from the top to the protilage side;
  • all three heights, carried out at the same time, shift at the center of the figure, creating an orthocenter, which, in the form of a tricutaneous body, can be either in the middle or in the middle;
  • median - a line that connects the vertex from the middle of the prolongation side;
  • The cross of the median is the point of its severity, which is located in the middle of the figure;
  • bisector - a line that runs from the top to the point of the crossbar on the prolong side, the point of the crossline of three bisectors is the center of the inscribed stake.

Forgive the truth about trikutniki

Trikutniki, like Vlasne, and all figures, show their peculiarities and power. As has already been said, this figure is the simplest one, but without its characteristic signs:

  • opposite the side found, lie next to the larger size, and on the contrary;
  • opposite the equal sides lie the equal cutlets, the butt of which is the equilateral tricutule;
  • the amount of internal circles will once again be equal to 180 °, which has already been demonstrated on the butt;
  • when one side of the tricut is extended, a new tunic is created between its boundaries, which is always related to the sum of tunics that are not adjacent to it;
  • Either side is always less than the sum of the other two sides, but their difference is greater.

Types of tricutniks

The next stage of getting to know each other is with the designated group, before the introduction of trikutnik. The belonging of one or another species depends on the size of the cuticles of the tricutaneous.

  • Equal thighs - with two equal sides, which are called hips, the third side of which acts as the basis of the figure. The bases of such a triangle are the same, and the median, drawn from the apex, is a bisector and height.
  • A correct, or equal-sided tricut is one in which all sides are equal.
  • Straight-cut: one of the cuts is equal to 90°. In this case, the side that lies opposite the side is called the hypotenuse, and the other two are called the legs.
  • State tricutnik - all kuti are less than 90 °.
  • Blunt - one of the corners greater than 90°.

Jealousy is like the trikutniks

In the process, it’s important to look closely at the taken figure, and to align the two three-pieces. And this, it would seem, is a simple topic with a lot of rules and theorems, behind which one can conclude that the figures that are being looked at are the trikulets. Signs of the jealousy of the trikutniki can be as follows: the trikutniki are equal, both on the same sides and on the same sides. If you put these two figures one on top of the other, all their lines will come together. Also, the figures can be similar, but at the same time there are practically the same figures, which differ in size. In order to develop such a summary about the presentation of the tricubitules, it is necessary to add one of the following minds:

  • two sides of one figure are comparable to two sides of another;
  • the two sides of one are proportional to the two sides of the other knit, and the sizes of the kuts created by the sides become similar;
  • three sides of the other figure are the same as in the first one.

Of course, for absolute fairness, so as not to raise any doubts, it is necessary to keep the same values ​​of all elements of both figures, it is important to say goodbye to the vicarious theorems, and to prove the fairness of the tricutaneous tolerances There are only a few minds available.

The first sign of jealousy of the trikutniki

The work on this topic revolves around the proof of a theorem that goes like this: “If the two sides of the tricubitus and the body that creates the stench are similar to the two sides and the coil of the other tricubitus, then these figures are also equal to each other.”

How does confirmation of the theorem about the first sign of jealousy of trikutniks sound? Everyone knows that the two sections of the river are equal, as they smell like the same thing, and the stakes of the river are similar, as they move in a new radius. And at the end of the three-pieces there is a small sign, which is visible, one can assume that the figures are identical, which is easy to correct in the course of the rise of various geometrical tasks.

What the theorem “The first sign of jealousy of the trikutniks” sounds like is described above, and the axis of the proof:

  • For example, the knitted pieces ABC and A 1 B 1 Z 1 are drawn on the same sides AB and A 1 B 1 and, obviously, BC and B 1 Z 1, and the knits that are made by these sides are drawn by the same size, then they are equal. Then, by pressing ABC on A A 1 B 1 C 1, you can select all lines and vertices. It turns out that the trivets are absolutely identical, and therefore equal to each other.

The theorem “The First Sign of Equality of Tricutniks” is also called “On Two Sides and a Cut.” Vlasne, in which lies the essence.

Theorem about the other sign

Another sign of zeal is presented in a similar way, the proof is based on the fact that when the figures are superimposed one on top of the other, the smells run together on all vertices and sides. And the theorem sounds like this: “If one side and two parts, in the knowledge of which they take part, resemble the sides and two parts of another tricupus, then these figures are identical, then they are equal.”

The third sign is proof

As 2, and 1 signs of jealousy of the three-pieces stand on both sides and sides of the figure, then the third one should be placed on the sides. Well, the theorem can be formulated as follows: “If all sides of one triangle are similar to three sides of another triangle, then the figures are identical.”

In order to complete this theorem, it is necessary to delve into the meaning of equality itself in more detail. Vlasna, what does the expression “tricutniks of the river” mean? Identity means that if you put one figure on top of another, all their elements are combined, but only if their sides are equal. At the same time there is a cut that lies on one side, like the same one, like another trikutnik, similar to the other top of the other figure. It should be noted that the proof here can easily be translated into 1 sign of the trikutniki’s jealousy. If such consistency is not guarded against, the jealousy of the tricutaneous people is simply impossible, due to these episodes, if the figure is the mirror image of the first.

Rectokutny trikutniki

Such tricuts have apexes with a cut of 90°. The following is true:

  • knitwear with a straight cut, because the legs of one are identical to the legs of the other;
  • the figures are equal, as they are equal to their hypotenuses and one of the legs;
  • These knitwear are equal, as their legs and sides are identical.

This sign is applied to the Proof of the theorem to create additional figures one by one, as a result of which the tricubitules are folded on the legs so that they have two straight ends with sides CA and CA 1.

More practical

Most often, it is practical to stagnate the first sign of jealousy of the trikutniki. In fact, it would seem that a simple topic for the 7th grade with geometry and planimetry is used to calculate the value of, for example, a telephone cable without any significant locality, which we pass through. With the help of this theorem, it is easy to develop the necessary structures for the purpose of the island, which is in the middle of the river, without flowing over it. Either use the parkan, stretch the plank in the spill so that it divides it into two equal tricuts, or open up the folding elements of the work in the carpentry, or when the blood system is broken up, I give it at the hour of waking up.

The first sign of the zeal of the trikutniks is widely observed in real “adult” life. Although at school, this very topic for the rich seems tedious and completely unnecessary.

Lesson summary

“The first sign of zeal of the trikutniks”

(Lesson No. 1, 7th grade, taught by Atanasyan L.S.)

Lesson objectives:

Navchalna:

Introduce the concept of theorem and proof of theorem;

To bring the sign of zeal to the trikutniks;

Learn to recognize the old stagnation of the first signs of jealousy of the tricutaneous.

Rozvivayucha:

Create a mental picture, highlight the focus, evaluate the flow of minds on the result;

Develop a more logical thinking for students.

Vikhovna:

Analyze the data, draw logical conclusions from these changes of mind, and work on the results;

Virbiti vminnya zoseredzhuvat vagu, zoseredzhuvat.

The meta is methodical: try a new approach before formulating a theorem, understand how to seize the learning moment when your mind becomes sufficient.

Lesson type: combinations.

Installation: computer, screen, projector, presentation, ruler, trikutnik,

color creds.

Lesson progress

Organizational moment: (2 xv)

At the beginning of the lesson, the leaders of the Trikutniki began their education. It was said that two figures, together with two trikulets, are called equal. Nowadays, it is clear that it is possible to establish the equality of two figures without actually superimposing one on the other, but by equalizing several elements of these figures, by squaring, as by aligning the triangles.

Review of the material covered: ( 6 xv)

Let's repeat the material from last lesson.

Theoretical food supply:

explain what kind of figure is called a triquite;

place the trikuti and show the sides of the top and the corner;

What is the perimeter of the tricutaneum?

What kind of trikutniki are called equal?

The skin can see an envelope, which has 6-7 paper's tricutaneous tissues; Students are encouraged to know that they have equal numbers.

When the search is completed, marry one of the students, as you know the pair. Learn how you put a trikutnik on another.

Vikonannya of a practical command with an offensive reversal:

No. 1: ∆DEK, ∆MNP are placed on the back (or slide).

Malyunok 1

Name the cut:

a) ∆DEK, which lies to the side ЄК;

b) ∆MNP, which lies up to the MN side.

Name the place:

a) ∆DEK, laying between sides DE and DK;

b) ∆MNP, placements between sides NP and RM.

Between all parties:

a) ∆DEK laid cut K;

b) ∆MNP laid cut N?

Malyunok 2

I call the teacher to the school, she accompanies her testimony with a demonstration in the chairs and a note on the school.

3. Introduction to new material: ( 16 xv)

To establish the equality of two tricuts, you need to connect them and check the equality of different sides and different parts. Six jealousies! However, there is no possibility of eating or verifying all six zealities. However, it is not necessary, it seems sufficient to install just a part of them. Our meta means which of these six jealousies are truly necessary.

Well, there's a problem.

Let's take those decisions.

Malyunok 3

There appears to be a fair assertion: “If two sides and a knitwear between them are obviously equal to two sides and a different knitwear between them, then such knitwear are equal.” This affirmation is called “The first sign of zeal of the trikutniks.”

And in mathematics there is a stricture, the justice of which is established by the way of merkuvan, is called a theorem, and the world itself is called proof of the theorem.

What theorems do we already know?

The power of the sizable kuti and the power of the vertical kuti.

Why is the theorem about the jealousy of trikutniks called a sign?

Sign ( after V. Dahl) - this is a sign, dignity, everything that is recognized. Having seen the frosty weather outside, you can say without leaving the house that it’s cold outside. To find out if the number 7859467 is divisible by 9, you don’t have to worry about divisibility: you can quickly use the divisibility sign.

The sign makes it possible to establish the equality of two tricuputnars without actually superimposing one of them on the other, but by equating the elements of the tricuputnars.

Whether the theorem is formed from the mind or the way it is laid out. How do you understand what can be meant by the phrase “mind of a theorem” and what by “laying out a theorem”?

Umov – these are already known facts about which the theorem is to be found, and again – these are the facts that need to be explained.

See the intellectual theorem “The first signs of jealousy of the Tricutniks.”

As the two sides of one triangle are similar to the two sides of another triangle.

See the summary of the theorem.

These are the trikutniks of equals.

Well, let me convey to you the sign of zeal of the trikutniks:

And now let's look at one more food. First of all, listen carefully to the formulation: If the two sides and the cut of one tricubitule are obviously equal to the two sides and the cut of another tricubitule, then the same tricubitules are equal. How do you respect what is the right thing to say?

Let's take a look at ∆ ABC and ∆ADC.

Malyunok 4

Side AB of the tricubitule ABC is older than side AD of the tricube ADC, side AC is the backside, and side C is the backside. Ale trikutniki are not equal. Well, the mind is hardened by vikonano, but the mind is not. This means that this is not true. Show special respect to those whose minds are indispensable!

4. Attaching new material: (10 xv)

Let's take a look at how you can put the theorem in perspective.

As soon as possible, the order for the prepared chairs, followed by the preparation of boards and slides on the back.

№2:

To enhance your skin care, I study up to school, commenting on the solutions, showing the riddles of the elements on the chair. The student's hearing is corrected, corrected, and additional evidence is provided where necessary.

I emphasize the respect of the scholars on the complexity of the local message “Trikutniks are equal on both sides and between them”, and not the formal “Tikuntniks are equal behind the first sign”, and I will be clear about everything. d decision, since the food was lost, I myself testify to them .

In the problem, how do you need to prove that the two three-pieces are equal, so that you can solve the problem by using a theorem or a theorem?

Well, theorem. It is necessary to strengthen the knitwear until the point is established, and, therefore, to verify the theorem, the three elements must be verified.

In Fig. AB = AC, 1 = 2.

a) Convey that the tricutaneous ABD and ACD are equal;

b) find BD and AB, if AC = 15 cm, DC = 5 cm.

Given: AB = AC, 1 = 2,

AC=15 div, DC=5 div.

Bring:

∆АВD = ∆АСD.

Know: ВD, АВ.

Brought to you by: First of all, you need to make decisions on your homework, teach your homework and learn how to complete your task. One scientist commentsFinished. The other one is the discovery of dovzhin vidrazkov. And thenWe write down the assigned task: I’m at home, studying at sewing.

Possible record of decision:

Brought to you by:

Let's take a look at ∆АВD and ∆АСD.

AB = AC (for minds)

АD – back side ∆АВD = ∆АСD (in two

1 = 2 (by definition) sides and a bundle between them)

Verbal commentary: the ABD and ACD jerseys are equal on both sides and between them, the first sign of the equality of the jerseys, which says: “As two sides and the circle between them of one tricubitron are equally friendly to the two sides and between them there is another trikutnik, then such trikutniks are equal ."

ВD = DC = 5 cm, АВ = AC = 15 cm.

Example: ВD = 5 cm, АВ = 15 cm.

Z'yasovaya, don't worry about food as the decision progresses.

5. Lesson tip:(4 xv)

Hey, let's repeat it:

How are trikutniks called equals?

What is called a theorem?

What is called proving a theorem?

How did we prove the theorem today? Formulate її.

Why is the theorem called a sign?

The teachings indicate nutrition.

I give grades for the work in the lesson with comments.

6. Home improvement: ( 2 xv)

P 15. Power supply 3-4 sides. 49-50. No. 93, 95.

No. 93. The cuts AE and DC are drawn at the point that is the middle of the skin of them. A) Convey that trikutniki ABC and EBD are equal; b) find the A and C cutoffs of the tricutulum ABC, since the tricumulus BDE has D = 470, E = 420.

No. 95. In Fig. BC = AD, 1 = 2, a) Show that ABC and CDA are equal; b) Find AB and BC, if AD = 17 cm, DC = 14 cm.

List of references:

Atanasyan L.S., Butuzov V.F. ta in. Geometry 7-9 grades. Handbook for 7-9 grades of secondary school. - M: Prosvitnitstvo, 2006.

Atanasyan L.S., Butuzov V.F. ta in. Teaching geometry in grades 7-9. Methodically approach your handyman. - M: Prosvitnitstvo, 2000.

Kovalova G.I., Mazurova N.I. Tests for in-line and remote control. Exhibition "Vchitel" 2008. .

Amelkin V.V., Rabtsevich T.I. School geometry in armchairs and formulas. 2008.

Theorem 3.1 (sign of jealousy of trikutniki on both sides and a fold between them). kut between them there is one trikutnik equal to two sides and between them there is another trikutnik, such trikutniks are equal.

Finished. Let ABC = A1B1C1A=A1, AB=A1B1, AC=A1C1 (Fig. 44). Let's see what tricutniks Rivni.

Let A 1 B 2 C 2 be a tricut, equal to the tricut ABC, with vertex B 2 on the exchange A 1 B 1 and vertex C 2 on the same plane along the straight line A 1 B 1 where vertex C 1 lies (Fig. 45, a) .

So since A 1 B 1 = A 1 B 2, then vertex B 2 meets vertex B 1 (Fig. 45.6). So since B 1 A 1 C 1 = B 2 A 1 C 2, then the exchange A 1 C 2 coincides with the exchange A 1 C 1 (Fig. 45, c). The fragments A 1 C 1 = A 1 C 2 then the vertex Z 2 runs behind the vertex C 1 (Fig. 45, d).

Well, the trikutnik A 1 B 1 C 1 runs together with the trikutnik A 1 B 2 C 2 means that it is similar to the trikutnik ABC. The theorem has been proven.



Zavdannya (1). The AB and CD sections move at point O, which is the middle of the skin of them. Why is section BD more important than section AC = 10 m?


Decision. Trikutniki AOS and BOD rivni behind the first sign Equality of Tricutniks(Fig. 46).

The parts AOS and BOD are equal to vertical, and OA=OB and OC=OD are the fragments of the point O in the middle of the sections AB and CD. From the equality of the tricutaneous AOS and BOD, the equality of their sides AC and BD is reflected. Shards for washing up AC = 10 m, then BD = 10 m.

A. V. Pogorelov, Geometry for grades 7-11, Handbook for backlighting installations

Two knitted pieces are called equal because they can be combined with overlays. For baby 1 there is an image of the tricutaneous ABC and A1B1C1. The skin of these tricutudies can be placed on another so that they are completely combined, so that their tops and sides are joined in pairs. It is clear that in this case, the tunics of these three-pieces come together in pairs.

In this way, since two trikulets are equal, then the elements (either sides or sides) of one trikuti are obviously similar to the elements of another trikuti. Please note that in equal trikutniki against similar equal sides(to join when applied) lie flat on the corners, and back: Lay the sides straight against the evenly cut sides.

So, for example, in the equal parts ABC and A 1 B 1 C 1, shown in figure 1, opposite the apparently equal sides AB and A 1 B 1 lie equal parts Z and C 1. The relationship between ABC and A1B1C1 is significant as follows: ΔABC = ΔA1B1C1. It turns out that the equality of the two trikulets can be established by equalizing the actions of their elements.

Theorem 1. The first sign of jealousy of the trikutniks. Since the two sides of one tricubitule are similar to the two sides of another tricubitule, then the same tricubitules are equal (Fig. 2).

Finished. Let's take a look at the three-pieces ABC і A 1 B 1 C 1, which are AB = A 1 B 1, AC = A 1 C 1 ∠ A = ∠ A 1 (div. Fig. 2). Let's see that ABC = A 1 B 1 C 1.

So since ∠ A = ∠ A 1, then the triangle ABC can be placed on the triangle A 1 B 1 C 1 so that the vertex A coincides with the vertex A 1, and the sides AB and AC are overlapped on the exchange A 1 B 1 and A 1 C 1 . The fragments AB = A 1 B 1, AC = A 1 C 1, then side AB is connected to the side A 1 B 1 and side AC is connected to the side A 1 C 1; zokrema, sum up the points i B 1, Z and C 1. So, the sides ZS and B1S1 are combined. Well, the trikutniks ABC I A 1 B 1 Z 1 are completely confused, well, they stink.

Theorem 2 can be proved in a similar way.

Theorem 2. Another sign of the trikutniks’ jealousy. If a side and two tufts of one tricut, which are adjacent to it, are similar to the equal sides and two tufts of another trikut, adjacent to it, then these are equal (Fig. 34).

Respect. Based on Theorem 2, Theorem 3 is established.

Theorem 3. The sum of any two internal cuts of the tricutaneous mensch for 180°.

The remaining theorem is followed by Theorem 4.

Theorem 4. The outer cut of the trikutnik is greater than any inner cut that is incompatible with it.

Theorem 5. The third sign of jealousy of the trikutniki. Since the three sides of one tricubitron are similar to the three sides of another tricubitron, then such tricubitrons are equal ().

butt 1. In the tricutanea ABC and DEF (Fig. 4)

∠ A = ∠ E, AB = 20 cm, AC = 18 cm, DE = 18 cm, EF = 20 cm. Trim ABC and DEF. What kind of kut does the DEF knit dress have?

Decision. These trikutniks are following the first sign. Cut F of the tricut DEF is similar to cut of ABC, since the cut lies opposite the parallel sides DE and AC.

butt 2. The sections AB and CD (Fig. 5) move at point O, which is the middle of the skin of them. Why is the BD section more expensive than the AC section than 6 m?

Decision. Trikutniki AOS and BOD level (behind the first sign): ∠ AOS = ∠ BOD (vertical), AB = OV, CO = OD (behind the washbasin).
The equality of these three-pieces reflects the equality of their sides, so AC = BD. If the fragments are behind the wash basin AC = 6 m, then BD = 6 m.

Tricutnik . Gostrokutny, blunt-cut and straight-cut trikutnik.

Cathetes and hypotenuse. Even-sided and even-sided knitwear.

Suma kutiv trikutnik.

The outer cut of the tricutaneous. Signs of jealousy of the tricutaneous people.

Miraculous lines and points at the tricutaneous: heights, medians,

bisectors, midpoints e perpendiculars, orthocenter,

center of gravity, center of stake, center of inscribed stake.

Pythagorean theorem. Spіvvіdshіndіsії іn іѕ іѕ іѕ іѕ іnѕtеr.

Tricutnik – this is a rich kutnik with three sides (or three sides). The sides of the trikutnik are often designated by small letters, which suggest to the great letters that they signify the prostrate peaks.

Since all three kuti are hot (Fig. 20), then gostrokutny trikutnik . Like one of the cuties is straight(C, Fig.21), then this straight cutter; sidesa, bwhat they do in a straight line is called legs; sidec, protilage to straight cut, called hypotenuse. I am one of kutiv tupii (B, Fig. 22), then this stupid tricut.


Tricutnik ABC (Fig. 23) - equiosceles, yakscho two yo side of the river (a= c); these equal parties are called scourge, the third party is called basis Tricutnik. Tricutnik ABC (Fig.24) - equilateral, yakscho Mustache yo side of the river (a = b = c). U zagalnym vipadku ( a ≠ b ≠ c) maєmo non-unilateral tricutnik .

The main authorities of the Trikutniks. For any trikutnik:

1. Opposite the larger side lies the larger kut, and by the way.

2. Opposite the equal sides lie equal parts, and on the contrary.

Zokrema, everyone is in equilateral trikutnik equal.

3. The amount of kutiv trikutnik is 180 º .

From the two remaining authorities it is clear that the skin of the equal side

trikutnik is older than 60 º.

4. Continue chewing one of the sides of the tricut (AC, Fig. 25), negated external

kut BCD . The outer cut of the trikutnik is equal to the amount of the inner cut,

not related to him : BCD = A + B.

5. Be-yaka the side of the jersey is smaller than the sum of the other two sides and more

Their differences (a < b + c, a > b – c;b < a + c, b > a – c;c < a + b,c > a – b).

Signs of jealousy of the tricutaneous people.

Trikutniki Rivna, as the stench is unique in Rivne:

a ) two sides and between them;

b ) two kuti and the side adjacent to them;

c) three sides.

Signs of jealousy of straight-cut tricutaneous.

Two straight-cut Knitted fabric of equals, as one of the advancing minds concludes:

1) equal parts of their legs;

2) the leg and hypotenuse of one tricube are equal to the leg and hypotenuse of the other;

3) the hypotenuse and the acute cut of one tricutaneous equal to the hypotenuse and the acute cut of the other;

4) the leg and the adjacent gostry cut of one tricube are comparable to the leg and the adjacent gostry cut of another;

5). to the protracted gastrointestinal tract of something else.

Miraculous lines and dots on the trikuputnik.

Height trikutnika - tseperpendicular,omissions from any top to the proximal side ( or else it's a continuation). This side is calledthe basis of the tricutaneous . The three heights of the trikutnik begin to shuffleat one pointcalled orthocenter Tricutnik. Orthocenter of the gostrocutaneous tricutule (point O , Fig. 26) of the lacerations in the middle of the tricubitule, andorthocenter of the obtuse cuticle (point O , Mal.27) – call; The orthocenter of the rectum tricutule converges with the apex of the rectum tricutule.

Median – tse video , which connects the top of the tricut from the middle of the prolong side. Three medians of the tricutaneous (AD, BE, CF, Fig. 28) shift at one point O first lie in the middle of the trikutnik and yes center of importance. At this point, divide the skin median 2:1, swelling at the top.

Bisector – tse section of bisector from the top to the point the webbing is on the pro-leg side. Three bisectors of the tricutaneous (AD, BE, CF, Fig. 29) shift at one point Oh, what will you ever lie in the middle of the trikutnik?і what? the center of the inscribed stake(Div. section “Inscribedthat describes the rich bushes").

A bisection divides the back side into parts proportional to the adjacent sides ; for example, in Fig. 29 AE: CE = AB: BC.

Middle perpendicular – center perpendicular, drawing from the middle cutting points (sides). Three perpendicular bisectors ABC(KO, MO, NO, Fig. 30 ) move at the same point Oh, what is it center the described stake (points K, M, N – the middle of the sides of the tricut ABC).

In the gostrocutaneous tricuputon, this point lies in the middle of the tricuputnum; for the slow-witted – zovni; y straight-cut - at the middle of the hypotenuse. Orthocenter, center of gravity, center of the described and center of the inscribed stake They are less likely to be avoided by the equal-sided tricut.

Pythagorean theorem. The straight-cut tricutaneous square has a dovzhiniHypotenuses are equal to the sum of squares and dovzhin cathetes.

The proof of the Pythagorean theorem is obvious from Fig. 31. Let's take a look at the straight-cut tricutnik ABC with legs a, b and the hypotenuse c.

Forgettable square AKMB , vikorist and hypotenuse AB yak bik. Thenprodovzhi side of straight-cut tricutaneous ABC So, let's take a square CDEF the side of which is more ancienta+b.Now it’s clear what the area of ​​a square is CDEF is older ( a+b) 2 . On the other hand, sir area of ​​ancient sum area four straight-cut tricutaneous and square AKMB, then

c 2 + 4 (ab / 2) = c 2 + 2 ab,

zvidsi,

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

and the rest is possible:

c 2 =a 2 + b 2 .

Spіvvіdshіndіsії іn іѕ іѕ іѕ іѕ іnѕtеr.

In a zagalny form (for a happy trikutnik) we have:

c 2 =a 2 + b 2 – 2ab· cos C,

de C – cut between sidesaі b .

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