Definition .

A circumscribed quadrilateral is a quadrilateral whose sides all touch the circle. In this case, the circle is said to be inscribed in a quadrilateral.

What properties does a circle inscribed in a quadrilateral have? When can a circle be inscribed in a quadrilateral? Where is the center of the inscribed circle?

Theorem 1.

A circle can be inscribed in a quadrilateral if and only if the sums of its opposite sides are equal.

A circle can be inscribed in a quadrilateral ABCD if

And vice versa, if the sums of the opposite sides of the quadrilateral are equal:

then a circle can be inscribed in quadrilateral ABCD.

Theorem 2.

The center of a circle inscribed in a quadrilateral is the point of intersection of its bisectors.

O is the intersection point of the bisectors of the quadrilateral ABCD.

AO, BO, CO, DO are the bisectors of the angles of the quadrilateral ABCD,

that is, ∠BAO=∠DAO, ∠ABO=∠CBO, etc.

3. The tangent points of the inscribed circle lying on the sides extending from one vertex are equidistant from this vertex.

AM=AN,

5. The area of ​​a quadrilateral is related to the radius of the circle inscribed in it by the formula

where p is the semi-perimeter of the quadrilateral.

Since the sums of the opposite sides of a circumscribed quadrilateral are equal, the semiperimeter is equal to any of the pairs of sums of opposite sides.

For example, for a quadrilateral ABCD p=AD+BC or p=AB+CD and

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Goals.

Educational. Creating conditions for successful mastery of the concept of the described quadrilateral, its properties, characteristics and mastering the skills to apply them in practice.

Developmental. Development of mathematical abilities, creation of conditions for the ability to generalize and apply forward and backward train of thought.

Educational. Cultivating a sense of beauty through the aesthetics of drawings, surprise at the unusual

decision, formation of organization, responsibility for the results of one’s work.

1. Study the definition of a circumscribed quadrilateral.

2. Prove the property of the sides of the circumscribed quadrilateral.

3. Introduce the duality of the properties of the sums of opposite sides and opposite angles of inscribed and circumscribed quadrilaterals.

4. To provide experience in the practical application of the considered theorems when solving problems.

5. Conduct initial monitoring of the level of assimilation of new material.

Equipment:

• computer, projector;
• textbook “Geometry. 10-11 grades” for general education. institutions: basic and profile. auto levels A.V. Pogorelov.

Software: Microsoft Word, Microsoft Power Point.

Using a computer when preparing a teacher for a lesson.

Using a standard Windows operating system program, the following were created for the lesson:

1. Presentation.
2. Tables.
3. Blueprints.
4. Handout.

Lesson Plan

Organizing time. (2 minutes.)

Checking homework. (5 minutes.)

Learning new material. (28 min.)

Independent work. (7 min.)

Homework.(1 min.)

Lesson summary. (2 minutes.)

During the classes

1. Organizational moment. Greetings. State the topic and purpose of the lesson. Record the date and topic of the lesson in your notebook.

2. Checking homework.

3. Studying new material.

Work on the concept of a circumscribed polygon.

Definition. The polygon is called described about a circle, if All his sides concern some circle.

Question. Which of the proposed polygons are described and which are not and why?

<Презентация. Слайд №2>

Proof of the properties of the circumscribed quadrilateral.

<Презентация. Слайд №3>

Theorem. In a circumscribed quadrilateral, the sums of opposite sides are equal.

Students work with the textbook and write down the formulation of the theorem in a notebook.

1. Present the formulation of the theorem in the form of a conditional sentence.

2. What is the condition of the theorem?

3. What is the conclusion of the theorem?

Answer. If a quadrilateral is circumscribed about a circle, That the sums of the opposite sides are equal.

The proof is carried out, students make notes in their notebooks.

<Презентация. Слайд №4>

Teacher. Note duality situations for sides and angles of circumscribed and inscribed quadrilaterals.

Consolidation of acquired knowledge.

Tasks.

The opposite sides of the described quadrilateral are 8 m and 12 m. Is it possible to find the perimeter?

Tasks based on finished drawings.<Презентация. Слайд №5>

Answer. 1. 10 m. 2. 20 m. 3. 21 m

Proof of the characteristic of a circumscribed quadrilateral.

State the converse theorem.

Answer. If in a quadrilateral the sums of opposite sides are equal, then a circle can be inscribed in it. (Return to slide 2, Fig. 7) <Презентация. Слайд №2>

Teacher. Clarify the formulation of the theorem.

Theorem. If the sums of opposite sides convex quadrilateral are equal, then a circle can be inscribed in it.

Working with the textbook. Get acquainted with the proof of the test for a circumscribed quadrilateral using the textbook.

Application of acquired knowledge.

3. Tasks based on finished drawings.

1. Is it possible to inscribe a circle in a quadrilateral with opposite sides 9 m and 4 m, 10 m and 3 m?

2. Is it possible to inscribe a circle into an isosceles trapezoid with bases of 1 m and 9 m, and a height of 3 m?

<Презентация. Слайд №6>

Written work in notebooks

.

Task. Find the radius of a circle inscribed in a rhombus with diagonals 6 m and 8 m.

<Презентация. Слайд № 7>

4. Independent work.

1 option

1. Is it possible to inscribe a circle

1) into a rectangle with sides 7 m and 10 m,

2. The opposite sides of a quadrilateral circumscribed about a circle are 7 m and 10 m.

Find the perimeter of the quadrilateral.

3. An equilateral trapezoid with bases 4 m and 16 m is described around a circle.

1) radius of the inscribed circle,

Option 2

1. Is it possible to inscribe a circle:

1) in a parallelogram with sides 6 m and 13 m,

2) squared?

2. The opposite sides of a quadrilateral circumscribed about a circle are 9 m and 11 m. Find the perimeter of the quadrilateral.

3. An equilateral trapezoid with a side side of 5 m is circumscribed about a circle with a radius of 2 m.

1) the base of the trapezoid,

2) radius of the circumscribed circle.

5. Homework. P.86, No. 28, 29, 30.

6. Lesson summary. Independent work is checked and grades are given.

<Презентация. Слайд № 8>

1 . The sum of the diagonals of a convex quadrilateral is greater than the sum of its two opposite sides.

2 . If the segments connecting the midpoints of opposite sides quadrilateral

a) are equal, then the diagonals of the quadrilateral are perpendicular;

b) are perpendicular, then the diagonals of the quadrilateral are equal.

3 . The bisectors of the angles on the lateral side of the trapezoid intersect at its midline.

4 . The sides of the parallelogram are equal and . Then the quadrilateral formed by the intersections of the bisectors of the angles of the parallelogram is a rectangle whose diagonals are equal to .

5 . If the sum of the angles at one of the bases of the trapezoid is 90°, then the segment connecting the midpoints of the bases of the trapezoid is equal to their half-difference.

6 . On the sides AB And AD parallelogram ABCD points taken M And N so straight MS And NC divide the parallelogram into three equal parts. Find MN, If BD=d.

7 . A straight line segment parallel to the bases of a trapezoid, enclosed inside the trapezoid, is divided by its diagonals into three parts. Then the segments adjacent to the sides are equal to each other.

8 . Through the point of intersection of the diagonals of the trapezoid with the bases, a straight line is drawn parallel to the bases. The segment of this line enclosed between the lateral sides of the trapezoid is equal to .

9 . A trapezoid is divided by a straight line parallel to its bases, equal to and , into two equal trapezoids. Then the segment of this line enclosed between the sides is equal to .

10 . If one of the following conditions is true, then the four points A, B, C And D lie on the same circle.

A) CAD=CBD= 90°.

b) points A And IN lie on one side of a straight line CD and angle CAD equal to angle CBD.

c) straight AC And BD intersect at a point ABOUT And O A OS=OV OD.

11 . Straight line connecting a point R intersection of the diagonals of a quadrilateral ABCD with dot Q line intersections AB And CD, divides the side AD in half. Then she divides in half and side Sun.

12 . Each side of a convex quadrilateral is divided into three equal parts. The corresponding division points on opposite sides are connected by segments. Then these segments divide each other into three equal parts.

13 . Two straight lines divide each of the two opposite sides of a convex quadrilateral into three equal parts. Then between these lines lies a third of the area of ​​the quadrilateral.

14 . If a circle can be inscribed in a quadrilateral, then the segment connecting the points at which the inscribed circle touches the opposite sides of the quadrilateral passes through the point of intersection of the diagonals.

15 . If the sums of the opposite sides of a quadrilateral are equal, then a circle can be inscribed in such a quadrilateral.

16. Properties of an inscribed quadrilateral with mutually perpendicular diagonals. Quadrangle ABCD inscribed in a circle of radius R. Its diagonals AC And BD mutually perpendicular and intersect at a point R. Then

a) median of a triangle ARV perpendicular to the side CD;

b) broken line AOC divides a quadrilateral ABCD into two equal-sized figures;

V) AB 2 +CD 2=4R 2 ;

G) AR 2 +BP 2 +CP 2 +DP 2 = 4R 2 and AB 2 +BC 2 +CD 2 +AD 2 =8R 2;

e) the distance from the center of the circle to the side of the quadrilateral is half the opposite side.

e) if the perpendiculars dropped to the side AD from the tops IN And WITH, cross the diagonals AC And BD at points E And F, That BCFE- rhombus;

g) a quadrilateral whose vertices are projections of a point R on the sides of the quadrilateral ABCD,- both inscribed and described;

h) a quadrilateral formed by tangents to the circumcircle of the quadrilateral ABCD, drawn at its vertices, can be inscribed in a circle.

17 . If a, b, c, d- successive sides of a quadrilateral, S is its area, then , and equality holds only for an inscribed quadrilateral whose diagonals are mutually perpendicular.

18 . Brahmagupta's formula. If the sides of a cyclic quadrilateral are equal a, b, c And d, then its area S can be calculated using the formula,

Where - semi-perimeter of a quadrilateral.

19 . If a quadrilateral with sides A, b, c, d can be inscribed and a circle can be described around it, then its area is equal to .

20 . Point P is located inside the square ABCD, and the angle PAB equal to angle RVA and is equal 15°. Then the triangle DPC- equilateral.

21 . If for a cyclic quadrilateral ABCD equality is satisfied CD=AD+BC, then the bisectors of its angles A And IN intersect on the side CD.

22 . Continuations of opposite sides AB And CD cyclic quadrilateral ABCD intersect at a point M, and the parties AD And Sun- at the point N. Then

a) angle bisectors AMD And D.N.C. mutually perpendicular;

b) straight MQ And NQ intersect the sides of the quadrilateral at the vertices of the rhombus;

c) point of intersection Q of these bisectors lies on the segment connecting the midpoints of the diagonals of the quadrilateral ABCD.

23 . Ptolemy's theorem. The sum of the products of two pairs of opposite sides of a cyclic quadrilateral is equal to the product of its diagonals.

24 . Newton's theorem. In any circumscribed quadrilateral, the midpoints of the diagonals and the center of the inscribed circle are located on the same straight line.

25 . Monge's theorem. Lines drawn through the midpoints of the sides of an inscribed quadrilateral perpendicular to the opposite sides intersect at one point.

27 . Four circles, built on the sides of a convex quadrilateral as diameters, cover the entire quadrilateral.

29 . Two opposite angles of a convex quadrilateral are obtuse. Then the diagonal connecting the vertices of these angles is less than the other diagonal.

30. The centers of squares built on the sides of a parallelogram outside it form a square themselves.

Material from Wikipedia - the free encyclopedia

• In Euclidean geometry, inscribed quadrilateral is a quadrilateral whose vertices all lie on the same circle. This circle is called circumscribed circle quadrilateral, and the vertices are said to lie on the same circle. The center of this circle and its radius are called respectively center And radius circumscribed circle. Other terms for this quadrilateral: a quadrilateral lies on one circle, the sides of the last quadrilateral are chords of the circle. A convex quadrilateral is usually assumed to be a convex quadrilateral. The formulas and properties given below are valid in the convex case.
• They say that if a circle can be drawn around a quadrilateral, That the quadrilateral is inscribed in this circle, and vice versa.

General criteria for the inscription of a quadrilateral

• Around a convex quadrilateral $\backslash pi$ radians), that is:

$\backslash angle\; A+\backslash angle\; C\; =\; \backslash angle\; B\; +\; \backslash angle\; D\; =\; 180^\backslash circ$

or in the figure notation:

$\backslash alpha\; +\; \backslash gamma\; =\; \backslash beta\; +\; \backslash delta\; =\; \backslash pi\; =\; 180^(\backslash circ).$

• It is possible to describe a circle around any quadrilateral in which the four perpendicular bisectors of its sides intersect at one point (or mediatrices of its sides, that is, perpendiculars to the sides passing through their midpoints).
• You can describe a circle around any quadrilateral that has one exterior angle adjacent to given internal angle, is exactly equal to the other interior angle opposite given internal corner. In essence, this condition is the condition of antiparallelism of two opposite sides of the quadrilateral. In Fig. Below is the outer and adjacent inner corners of a green pentagon.

$\backslash displaystyle\; AX\backslash cdot\; XC\; =\; BX\backslash cdot\; XD.$

• Intersection X may be internal or external to the circle. In the first case, we obtain the cyclic quadrilateral is ABCD, and in the latter case we obtain an inscribed quadrilateral ABDC. When intersecting inside a circle, the equality states that the product of the lengths of the segments in which the point X divides one diagonal, is equal to the product of the lengths of the segments in which the point X divides another diagonal. This condition is known as the "intersecting chord theorem." In our case, the diagonals of the inscribed quadrilateral are the chords of the circle.
• Another criterion for inclusion. Convex quadrilateral ABCD a circle is inscribed if and only if

$\backslash tan(\backslash frac(\backslash alpha)(2))\backslash tan(\backslash frac(\backslash gamma)(2))=\backslash tan(\backslash frac(\backslash beta)(2))\backslash tan(\backslash frac(\backslash delta)(\; 2))=1.$

Particular criteria for the inscription of a quadrilateral

A simple inscribed (without self-intersection) quadrilateral is convex. A circle can be described around a convex quadrilateral if and only if the sum of its opposite angles is equal to 180° ( $\backslash pi$ radian). You can describe a circle around:

• any antiparallelogram
• any rectangle (a special case is a square)
• any isosceles trapezoid
• any quadrilateral that has two opposite right angles.

Properties

Formulas with diagonals

$ef=ac+bd$; $\backslash frac(e)(f)\; =\; \backslash frac(a\backslash cdot\; d+b\backslash cdot\; c)(a\backslash cdot\; b+c\backslash cdot\; d).$

In the last formula of the pair of adjacent sides of the numerator a And d, b And c rest their ends on a diagonal length e. A similar statement holds for the denominator.

• Formulas for diagonal lengths(consequences ):

$e\; =\; \backslash sqrt(\backslash frac((ac+bd)(ad+bc))(ab+cd))$ And $f\; =\; \backslash sqrt(\backslash frac((ac+bd)(ab+cd))(ad+bc))$

Formulas with angles

For a cyclic quadrilateral with a sequence of sides a , b , c , d, with semi-perimeter p and angle A between the parties a And d, trigonometric angle functions A are given by formulas

$\backslash cos\; A\; =\; \backslash frac(a^2\; +\; d^2\; -\; b^2\; -\; c^2)(2(ad\; +\; bc)),$ $\backslash sin\; A\; =\; \backslash frac(2\backslash sqrt((p-a)(p-b)(p-c)(p-d)))((ad+bc)),$ $\backslash tan\; \backslash frac(A)(2)\; =\; \backslash sqrt(\backslash frac((p-a)(p-d))((p-b)(p-c))).$

Corner θ between the diagonals there is:p.26

$\backslash tan\; \backslash frac(\backslash theta)(2)\; =\; \backslash sqrt(\backslash frac((p-b)(p-d))((p-a)(p-c))).$

• If opposite sides a And c intersect at an angle φ , then it is equal

$\backslash cos(\backslash frac(\backslash varphi)(2))=\backslash sqrt(\backslash frac((p-b)(p-d)(b+d)^2)((ab+cd)(ad+bc))),$

Where p there is a semi-perimeter. :p.31

Radius of a circle circumscribed about a quadrilateral

Parameshvara Formula

If a quadrilateral with consecutive sides a , b , c , d and semi-perimeter p inscribed in a circle, then its radius is equal to Parameshwar's formula:p. 84

$R=\; \backslash frac(1)(4)\; \backslash sqrt(\backslash frac((ab+cd)(ad+bc)(ac+bd))((p-a)(p-b)(p-c)(p-d))).$

It was derived by the Indian mathematician Parameshwar in the 15th century (c. 1380–1460)

• Convex quadrilateral (see figure on the right) formed by four data Mikel's straight lines, is inscribed in a circle if and only if the Mikel point M of a quadrilateral lies on a line connecting two of the six points of intersection of the lines (those that are not vertices of the quadrilateral). That is, when M lies on E.F..

A criterion that a quadrilateral composed of two triangles is inscribed in a certain circle

$f^2\; =\; \backslash frac((ac+bd)(ad+bc))((ab+cd)).$

• The last condition gives the expression for the diagonal f a quadrilateral inscribed in a circle through the lengths of its four sides ( a, b, c, d). This formula immediately follows when multiplying and when equating to each other the left and right parts of formulas expressing the essence Ptolemy's first and second theorems(see above).

A criterion that a quadrilateral cut by a straight line from a triangle is inscribed in a certain circle

• A straight line, antiparallel to the side of the triangle and intersecting it, cuts off a quadrilateral from it, around which a circle can always be described.
• Consequence. Around an antiparallelogram, in which two opposite sides are antiparallel, it is always possible to describe a circle.

Area of ​​a quadrilateral inscribed in a circle

Variations of Brahmagupta's formula

$S=\backslash sqrt((p-a)(p-b)(p-c)(p-d)),$ where p is the semi-perimeter of the quadrilateral. $S=\; \backslash frac(1)(4)\; \backslash sqrt(-\; \backslash begin(vmatrix)$

a & b & c & -d \\ b & a & -d & c \\ c & -d & a & b \\ -d & c & b & a \end(vmatrix))

Other area formulas

$S\; =\; \backslash tfrac(1)(2)(ab+cd)\backslash sin(B)$ $S\; =\; \backslash tfrac(1)(2)(ac+bd)\backslash sin(\backslash theta),$

Where θ any of the angles between the diagonals. Provided that the angle A is not a straight line, area can also be expressed as :p.26

$S\; =\; \backslash tfrac(1)(4)(a^2-b^2-c^2+d^2)\backslash tan(A).$ $\backslash displaystyle\; S=2R^2\backslash sin(A)\backslash sin(B)\backslash sin(\backslash theta),$

Where R is the radius of the circumcircle. As a direct consequence we have the inequality

$S\backslash le\; 2R^2,$

where equality is possible if and only if this quadrilateral is a square.

Brahmagupta quadrangles

Brahmagupta Quadrangle is a quadrilateral inscribed in a circle with integer side lengths, integer diagonals and integer area. All possible Brahmagupta quadrilaterals with sides a , b , c , d, with diagonals e , f, with area S, and the radius of the circumscribed circle R can be obtained by removing the denominators of the following expressions involving rational parameters t , u, And v :

$a=$ $b=(1+u^2)(v-t)(1+tv)$ $c=t(1+u^2)(1+v^2)$ $d=(1+v^2)(u-t)(1+tu)$ $e=u(1+t^2)(1+v^2)$ $f=v(1+t^2)(1+u^2)$ $S=uv$ $4R=(1+u^2)(1+v^2)(1+t^2).$

Examples

• Particular quadrilaterals inscribed in a circle are: rectangle, square, isosceles or isosceles trapezoid, antiparallelogram.

Quadrilaterals inscribed in a circle with perpendicular diagonals (inscribed orthodiagonal quadrilaterals)

Properties of quadrilaterals inscribed in a circle with perpendicular diagonals

Circumradius and area

For a quadrilateral inscribed in a circle with perpendicular diagonals, suppose that the intersection of the diagonals divides one diagonal into segments of length p 1 and p 2, and divides the other diagonal into length segments q 1 and q 2. Then (The first equality is Proposition 11 of Archimedes" Book of Lemmas)

$D^2=p\_1^2+p\_2^2+q\_1^2+q\_2^2=a^2+c^2=b^2+d^2,$

Where D- diameter of the circle. This is true because the diagonals are perpendicular to the chord of the circle. From these equations it follows that the radius of the circumscribed circle R can be written as

$R=\backslash tfrac(1)(2)\backslash sqrt(p\_1^2+p\_2^2+q\_1^2+q\_2^2)$

or in terms of the sides of a quadrilateral in the form

$R=\backslash tfrac(1)(2)\backslash sqrt(a^2+c^2)=\backslash tfrac(1)(2)\backslash sqrt(b^2+d^2).$

It also follows that

$a^2+b^2+c^2+d^2=8R^2.$

• For inscribed ordiagonal quadrilaterals, Brahmagupta's theorem holds:

If a cyclic quadrilateral has perpendicular diagonals intersecting at a point $M$, then two pairs of it antimediatris pass through a point $M$.

Comment. In this theorem under anti-mediatrix understand the segment $F.E.$ quadrilateral in the figure on the right (by analogy with the perpendicular bisector (mediatrix) to the side of the triangle). It is perpendicular to one side and at the same time passes through the middle of the opposite side of the quadrilateral.

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Notes

1. Bradley, Christopher J. (2007), The Algebra of Geometry: Cartesian, Areal and Projective Co-Ordinates,Highperception, p. 179, ISBN 1906338000, OCLC
2. . Inscribed quadrilaterals.
3. Siddons, A. W. & Hughes, R. T. (1929) Trigonometry, Cambridge University Press, p. 202, O.C.L.C.
4. Durell, C. V. & Robson, A. (2003), , Courier Dover, ISBN 978-0-486-43229-8 ,
5. Alsina, Claudi & Nelsen, Roger B. (2007), "", Forum Geometricorum T. 7: 147–9 ,
6. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007 (orig. 1929).
7. Hoehn, Larry (March 2000), "Circumradius of a cyclic quadrilateral", Mathematical Gazette T. 84 (499): 69–70
8. .
9. Altshiller-Court, Nathan (2007), College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle(2nd ed.), Courier Dover, pp. 131, 137–8, ISBN 978-0-486-45805-2, OCLC
10. Honsberger, Ross (1995), , Episodes in Nineteenth and Twentieth Century Euclidean Geometry, vol. 37, New Mathematical Library, Cambridge University Press, pp. 35–39, ISBN 978-0-88385-639-0
11. Weisstein, Eric W.(English) on the Wolfram MathWorld website.
12. Bradley, Christopher (2011), ,
13. .
14. Coxeter, Harold Scott MacDonald & Greitzer, Samuel L. (1967), , Geometry Revisited, Mathematical Association of America, pp. 57, 60, ISBN 978-0-88385-619-2
15. .
16. Andreescu, Titu & Enescu, Bogdan (2004), , Mathematical Olympiad Treasures, Springer, pp. 44–46, 50, ISBN 978-0-8176-4305-8
17. .
18. Buchholz, R. H. & MacDougall, J. A. (1999), "", Bulletin of the Australian Mathematical Society T. 59 (2): 263–9 , DOI 10.1017/S0004972700032883
19. .
20. Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ. Co., 2007
21. , With. 74.
22. .
23. .
24. .
25. Peter, Thomas (September 2003), "Maximizing the area of ​​a quadrilateral", The College Mathematics Journal T. 34 (4): 315–6
26. Prasolov, Viktor, ,
27. Alsina, Claudi & Nelsen, Roger (2009), , , Mathematical Association of America, p. 64, ISBN 978-0-88385-342-9 ,
28. Sastry, K.R.S. (2002). "" (PDF). Forum Geometricorum 2 : 167–173.
29. Posamentier, Alfred S. & Salkind, Charles T. (1970), , Challenging Problems in Geometry(2nd ed.), Courier Dover, pp. 104–5, ISBN 978-0-486-69154-1
30. .
31. .
32. .

see also

By number of sides Correct Triangles Quadrilaterals see also

The article contains short (“Harvard”) references to publications that are not listed or incorrectly described in the bibliographic section. List of broken links: , , , , , , , , , – Well, what, my Cossack? (Marya Dmitrievna called Natasha a Cossack) - she said, caressing Natasha with her hand, who approached her hand without fear and cheerfully. – I know that the potion is a girl, but I love her. She took out pear-shaped yakhon earrings from her huge reticule and, giving them to Natasha, who was beaming and blushing for her birthday, immediately turned away from her and turned to Pierre. - Eh, eh! kind! “Come here,” she said in a feignedly quiet and thin voice. - Come on, my dear... And she menacingly rolled up her sleeves even higher. Pierre approached, naively looking at her through his glasses. - Come, come, my dear! I was the only one who told your father the truth when he had a chance, but God commands it to you. She paused. Everyone was silent, waiting for what would happen, and feeling that there was only a preface. - Good, nothing to say! good boy!... The father is lying on his bed, and he is amusing himself, putting the policeman on a bear. It's a shame, father, it's a shame! It would be better to go to war. She turned away and offered her hand to the count, who could hardly restrain himself from laughing. - Well, come to the table, I have tea, is it time? - said Marya Dmitrievna. The count walked ahead with Marya Dmitrievna; then the countess, who was led by a hussar colonel, the right person with whom Nikolai was supposed to catch up with the regiment. Anna Mikhailovna - with Shinshin. Berg shook hands with Vera. A smiling Julie Karagina went with Nikolai to the table. Behind them came other couples, stretching across the entire hall, and behind them, one by one, were children, tutors and governesses. The waiters began to stir, the chairs rattled, music began to play in the choir, and the guests took their seats. The sounds of the count's home music were replaced by the sounds of knives and forks, the chatter of guests, and the quiet steps of waiters. At one end of the table the countess sat at the head. On the right is Marya Dmitrievna, on the left is Anna Mikhailovna and other guests. At the other end sat the count, on the left the hussar colonel, on the right Shinshin and other male guests. On one side of the long table are older young people: Vera next to Berg, Pierre next to Boris; on the other hand - children, tutors and governesses. From behind the crystal, bottles and vases of fruit, the Count looked at his wife and her tall cap with blue ribbons and diligently poured wine for his neighbors, not forgetting himself. The countess also, from behind the pineapples, not forgetting her duties as a housewife, cast significant glances at her husband, whose bald head and face, it seemed to her, were more sharply different from his gray hair in their redness. There was a steady babble on the ladies' end; in the men's room, voices were heard louder and louder, especially the hussar colonel, who ate and drank so much, blushing more and more, that the count was already setting him up as an example to the other guests. Berg, with a gentle smile, spoke to Vera that love is not an earthly, but a heavenly feeling. Boris named his new friend Pierre the guests at the table and exchanged glances with Natasha, who was sitting opposite him. Pierre spoke little, looked at new faces and ate a lot. Starting from two soups, from which he chose a la tortue, [turtle,] and kulebyaki and to hazel grouse, he did not miss a single dish and not a single wine, which the butler mysteriously stuck out in a bottle wrapped in a napkin from behind his neighbor’s shoulder, saying or “drey Madeira", or "Hungarian", or "Rhine wine". He placed the first of the four crystal glasses with the count's monogram that stood in front of each device, and drank with pleasure, looking at the guests with an increasingly pleasant expression. Natasha, sitting opposite him, looked at Boris the way thirteen-year-old girls look at a boy with whom they had just kissed for the first time and with whom they are in love. This same look of hers sometimes turned to Pierre, and under the gaze of this funny, lively girl he wanted to laugh himself, not knowing why. Nikolai sat far from Sonya, next to Julie Karagina, and again with the same involuntary smile he spoke to her. Sonya smiled grandly, but apparently was tormented by jealousy: she turned pale, then blushed and listened with all her might to what Nikolai and Julie were saying to each other. The governess looked around restlessly, as if preparing to fight back if anyone decided to offend the children. The German tutor tried to memorize all kinds of dishes, desserts and wines in order to describe everything in detail in a letter to his family in Germany, and was very offended by the fact that the butler, with a bottle wrapped in a napkin, carried him around. The German frowned, tried to show that he did not want to receive this wine, but was offended because no one wanted to understand that he needed the wine not to quench his thirst, not out of greed, but out of conscientious curiosity. At the male end of the table the conversation became more and more animated. The colonel said that the manifesto declaring war had already been published in St. Petersburg and that the copy that he himself had seen had now been delivered by courier to the commander-in-chief. - And why is it difficult for us to fight Bonaparte? - said Shinshin. – II a deja rabattu le caquet a l "Autriche. Je crins, que cette fois ce ne soit notre tour. [He has already knocked down the arrogance of Austria. I am afraid that our turn would not come now.] The colonel was a stocky, tall and sanguine German, obviously a servant and a patriot. He was offended by Shinshin's words. “And then, we are a good sovereign,” he said, pronouncing e instead of e and ъ instead of ь. “Then that the emperor knows this. He said in his manifesto that he can look indifferently at the dangers threatening Russia, and that the safety of the empire, its dignity and the sanctity of its alliances,” he said, for some reason especially emphasizing the word “unions”, as if this was the whole essence of the matter. And with his characteristic infallible, official memory, he repeated the opening words of the manifesto... “and the desire, the sole and indispensable goal of the sovereign: to establish peace in Europe on solid foundations - they decided to now send part of the army abroad and make new efforts to achieve this intention “. “That’s why, we are a good sovereign,” he concluded, edifyingly drinking a glass of wine and looking back at the count for encouragement. – Connaissez vous le proverbe: [You know the proverb:] “Erema, Erema, you should sit at home, sharpen your spindles,” said Shinshin, wincing and smiling. – Cela nous convient a merveille. [This comes in handy for us.] Why Suvorov - they chopped him up, a plate couture, [on his head,] and where are our Suvorovs now? Je vous demande un peu, [I ask you,] - he said, constantly jumping from Russian to French. “We must fight until the last drop of blood,” said the colonel, hitting the table, “and die for our emperor, and then everything will be fine.” And to argue as much as possible (he especially drew out his voice on the word “possible”), as little as possible,” he finished, again turning to the count. “That’s how we judge the old hussars, that’s all.” How do you judge, young man and young hussar? - he added, turning to Nikolai, who, having heard that it was about war, left his interlocutor and looked with all his eyes and listened with all his ears to the colonel. “I completely agree with you,” answered Nikolai, all flushed, spinning the plate and rearranging the glasses with such a decisive and desperate look, as if at the moment he was exposed to great danger, “I am convinced that the Russians must die or win,” he said. feeling the same way as others, after the word had already been said, that it was too enthusiastic and pompous for the present occasion and therefore awkward. “C"est bien beau ce que vous venez de dire, [Wonderful! What you said is wonderful],” said Julie, who was sitting next to him, sighing. Sonya trembled all over and blushed to the ears, behind the ears and to the neck and shoulders, in While Nikolai was speaking, Pierre listened to the colonel's speeches and nodded his head approvingly. “That’s nice,” he said. “A real hussar, young man,” shouted the colonel, hitting the table again. -What are you making noise about there? – Marya Dmitrievna’s bass voice was suddenly heard across the table. -Why are you knocking on the table? - she turned to the hussar, - who are you getting excited about? right, you think that the French are in front of you? “I’m telling the truth,” said the hussar, smiling. “Everything about the war,” the count shouted across the table. - After all, my son is coming, Marya Dmitrievna, my son is coming. - And I have four sons in the army, but I don’t bother. Everything is God’s will: you will die lying on the stove, and in battle God will have mercy,” Marya Dmitrievna’s thick voice sounded without any effort from the other end of the table. - This is true. And the conversation focused again - the ladies at their end of the table, the men at his. “But you won’t ask,” said the little brother to Natasha, “but you won’t ask!” “I’ll ask,” Natasha answered. Her face suddenly flushed, expressing desperate and cheerful determination. She stood up, inviting Pierre, who was sitting opposite her, to listen, and turned to her mother: - Mother! – her childish, chesty voice sounded across the table. - What do you want? – the countess asked in fear, but, seeing from her daughter’s face that it was a prank, she sternly waved her hand, making a threatening and negative gesture with her head. The conversation died down. - Mother! what kind of cake will it be? – Natasha’s voice sounded even more decisively, without breaking down. The Countess wanted to frown, but could not. Marya Dmitrievna shook her thick finger. “Cossack,” she said threateningly. Most of the guests looked at the elders, not knowing how to take this trick. - Here I am! - said the countess. - Mother! what kind of cake will there be? - Natasha shouted now boldly and capriciously cheerfully, confident in advance that her prank would be well received. Sonya and fat Petya were hiding from laughter. “That’s why I asked,” Natasha whispered to her little brother and Pierre, whom she looked at again. “Ice cream, but they won’t give it to you,” said Marya Dmitrievna. Natasha saw that there was nothing to be afraid of, and therefore she was not afraid of Marya Dmitrievna. - Marya Dmitrievna? what ice cream! I don't like cream. - Carrot. - No, which one? Marya Dmitrievna, which one? – she almost screamed. - I want to know! Marya Dmitrievna and the Countess laughed, and all the guests followed them. Everyone laughed not at Marya Dmitrievna’s answer, but at the incomprehensible courage and dexterity of this girl, who knew how and dared to treat Marya Dmitrievna like that. Natasha fell behind only when she was told that there would be pineapple. Champagne was served before the ice cream. The music started playing again, the count kissed the countess, and the guests stood up and congratulated the countess, clinking glasses across the table with the count, the children, and each other. Waiters ran in again, chairs rattled, and in the same order, but with redder faces, the guests returned to the drawing room and the count’s office. The Boston tables were moved apart, the parties were drawn up, and the Count's guests settled in two living rooms, a sofa room and a library. The Count, fanning out his cards, could hardly resist the habit of an afternoon nap and laughed at everything. The youth, incited by the countess, gathered around the clavichord and harp. Julie was the first, at the request of everyone, to play a piece with variations on the harp and, together with other girls, began to ask Natasha and Nikolai, known for their musicality, to sing something. Natasha, who was addressed as a big girl, was apparently very proud of this, but at the same time she was timid. - What are we going to sing? – she asked. “The key,” answered Nikolai. - Well, let's hurry up. Boris, come here,” Natasha said. - Where is Sonya? She looked around and, seeing that her friend was not in the room, ran after her. Running into Sonya’s room and not finding her friend there, Natasha ran into the nursery - and Sonya was not there. Natasha realized that Sonya was in the corridor on the chest. The chest in the corridor was the place of sorrows of the younger female generation of the Rostov house. Indeed, Sonya in her airy pink dress, crushing it, lay face down on her nanny’s dirty striped feather bed, on the chest and, covering her face with her fingers, cried bitterly, shaking her bare shoulders. Natasha's face, animated, with a birthday all day, suddenly changed: her eyes stopped, then her wide neck shuddered, the corners of her lips drooped. - Sonya! what are you?... What, what's wrong with you? Wow wow!… And Natasha, opening her big mouth and becoming completely stupid, began to roar like a child, not knowing the reason and only because Sonya was crying. Sonya wanted to raise her head, wanted to answer, but she couldn’t and hid even more. Natasha cried, sitting down on the blue feather bed and hugging her friend. Having gathered her strength, Sonya stood up, began to wipe away her tears and tell the story. - Nikolenka is leaving in a week, his... paper... came out... he told me himself... Yes, I still wouldn’t cry... (she showed the piece of paper she was holding in her hand: it was poetry written by Nikolai) I still wouldn’t cry, but you didn’t you can... no one can understand... what kind of soul he has. And she again began to cry because his soul was so good. “You feel good... I don’t envy you... I love you, and Boris too,” she said, gathering a little strength, “he’s cute... there are no obstacles for you.” And Nikolai is my cousin... I need... the metropolitan himself... and that’s impossible. And then, if mamma... (Sonya considered the countess and called her mother), she will say that I am ruining Nikolai’s career, I have no heart, that I am ungrateful, but really... for God’s sake... (she crossed herself) I love her so much too , and all of you, only Vera... For what? What did I do to her? I am so grateful to you that I would be glad to sacrifice everything, but I have nothing... Sonya could no longer speak and again hid her head in her hands and the feather bed. Natasha began to calm down, but her face showed that she understood the importance of her friend’s grief. - Sonya! - she said suddenly, as if she had guessed the real reason for her cousin’s grief. – That’s right, Vera talked to you after dinner? Yes? – Yes, Nikolai himself wrote these poems, and I copied others; She found them on my table and said that she would show them to mamma, and also said that I was ungrateful, that mamma would never allow him to marry me, and he would marry Julie. You see how he is with her all day... Natasha! For what?… And again she cried more bitterly than before. Natasha lifted her up, hugged her and, smiling through her tears, began to calm her down. - Sonya, don’t believe her, darling, don’t believe her. Do you remember how all three of us talked with Nikolenka in the sofa room; remember after dinner? After all, we decided everything how it would be. I don’t remember how, but you remember how everything was good and everything was possible. Uncle Shinshin’s brother is married to a cousin, and we are second cousins. And Boris said that this is very possible. You know, I told him everything. And he is so smart and so good,” Natasha said... “You, Sonya, don’t cry, my dear darling, Sonya.” - And she kissed her, laughing. - Faith is evil, God bless her! But everything will be fine, and she won’t tell mamma; Nikolenka will say it himself, and he didn’t even think about Julie. And she kissed her on the head. Sonya stood up, and the kitten perked up, his eyes sparkled, and he seemed ready to wave his tail, jump on his soft paws and play with the ball again, as was proper for him. - You think? Right? By God? – she said, quickly straightening her dress and hair. - Really, by God! – Natasha answered, straightening a stray strand of coarse hair under her friend’s braid. And they both laughed. - Well, let's go sing "The Key." - Let's go to. “You know, this fat Pierre who was sitting opposite me is so funny!” – Natasha suddenly said, stopping. - I'm having a lot of fun! And Natasha ran down the corridor. Sonya, shaking off the fluff and hiding the poems in her bosom, to her neck with protruding chest bones, with light, cheerful steps, with a flushed face, ran after Natasha along the corridor to the sofa. At the request of the guests, the young people sang the “Key” quartet, which everyone really liked; then Nikolai sang the song he had learned again. On a pleasant night, in the moonlight, Imagine yourself happily That there is still someone in the world, Who thinks about you too! As she, with her beautiful hand, Walking along the golden harp, With its passionate harmony Calling to itself, calling you! Another day or two, and heaven will come... But ah! your friend won't live! And he had not yet finished singing the last words when the young people in the hall were preparing to dance and the musicians in the choir began to knock their feet and cough. Pierre was sitting in the living room, where Shinshin, as if with a visitor from abroad, began a political conversation with him that was boring for Pierre, to which others joined. When the music started playing, Natasha entered the living room and, going straight to Pierre, laughing and blushing, said: - Mom told me to ask you to dance. “I’m afraid of confusing the figures,” said Pierre, “but if you want to be my teacher...” And he offered his thick hand, lowering it low, to the thin girl. While the couples were settling down and the musicians were lining up, Pierre sat down with his little lady. Natasha was completely happy; she danced with a big one, with someone who came from abroad. She sat in front of everyone and talked to him like a big girl. She had a fan in her hand, which one young lady had given her to hold. And, assuming the most secular pose (God knows where and when she learned this), she, fanning herself and smiling through the fan, spoke to her gentleman. - What is it, what is it? Look, look,” said the old countess, passing through the hall and pointing at Natasha. Natasha blushed and laughed. - Well, what about you, mom? Well, what kind of hunt are you looking for? What's surprising here? In the middle of the third eco-session, the chairs in the living room, where the count and Marya Dmitrievna were playing, began to move, and most of the honored guests and old people, stretching after a long sitting and putting wallets and purses in their pockets, walked out the doors of the hall. Marya Dmitrievna walked ahead with the count - both with cheerful faces. The Count, with playful politeness, like a ballet, offered his rounded hand to Marya Dmitrievna. He straightened up, and his face lit up with a particularly brave, sly smile, and as soon as the last figure of the ecosaise was danced, he clapped his hands to the musicians and shouted to the choir, addressing the first violin: - Semyon! Do you know Danila Kupor? This was the count's favorite dance, danced by him in his youth. (Danilo Kupor was actually one figure of the Angles.) “Look at dad,” Natasha shouted to the whole hall (completely forgetting that she was dancing with a big one), bending her curly head to her knees and bursting into her ringing laughter throughout the hall. Indeed, everyone in the hall looked with a smile of joy at the cheerful old man, who, next to his dignified lady, Marya Dmitrievna, who was taller than him, rounded his arms, shaking them in time, straightened his shoulders, twisted his legs, slightly stamping his feet, and with a more and more blooming smile on his round face, he prepared the audience for what was to come. As soon as the cheerful, defiant sounds of Danila Kupor, similar to a cheerful chatterbox, were heard, all the doors of the hall were suddenly filled with men's faces on one side and women's smiling faces of servants on the other, who came out to look at the merry master. - Father is ours! Eagle! – the nanny said loudly from one door. The count danced well and knew it, but his lady did not know how and did not want to dance well. Her huge body stood upright with her powerful arms hanging down (she handed the reticule to the Countess); only her stern but beautiful face danced. What was expressed in the count's entire round figure, in Marya Dmitrievna was expressed only in an increasingly smiling face and a twitching nose. But if the count, becoming more and more dissatisfied, captivated the audience with the surprise of deft twists and light jumps of his soft legs, Marya Dmitrievna, with the slightest zeal in moving her shoulders or rounding her arms in turns and stamping, made no less an impression on merit, which everyone appreciated her obesity and ever-present severity. The dance became more and more animated. The counterparts could not attract attention to themselves for a minute and did not even try to do so. Everything was occupied by the count and Marya Dmitrievna. Natasha pulled the sleeves and dresses of all those present, who were already keeping their eyes on the dancers, and demanded that they look at daddy. During the intervals of the dance, the Count took a deep breath, waved and shouted to the musicians to play quickly. Quicker, quicker and quicker, faster and faster and faster, the count unfolded, now on tiptoes, now on heels, rushing around Marya Dmitrievna and, finally, turning his lady to her place, made the last step, raising his soft leg up from behind, bending his sweaty head with a smiling face and roundly waving his right hand amid the roar of applause and laughter, especially from Natasha. Both dancers stopped, panting heavily and wiping themselves with cambric handkerchiefs. “This is how they danced in our time, ma chere,” said the count. - Oh yes Danila Kupor! - Marya Dmitrievna said, letting out the spirit heavily and for a long time, rolling up her sleeves. While the Rostovs were dancing the sixth anglaise in the hall to the sounds of tired musicians out of tune, and tired waiters and cooks were preparing dinner, the sixth blow struck Count Bezukhy. The doctors declared that there was no hope of recovery; the patient was given silent confession and communion; They were making preparations for the unction, and in the house there was the bustle and anxiety of expectation, common at such moments. Outside the house, behind the gates, undertakers crowded, hiding from the approaching carriages, awaiting a rich order for the count's funeral. The Commander-in-Chief of Moscow, who constantly sent adjutants to inquire about the Count’s position, that evening himself came to say goodbye to the famous Catherine’s nobleman, Count Bezukhim. The magnificent reception room was full. Everyone stood up respectfully when the commander-in-chief, having been alone with the patient for about half an hour, came out of there, slightly returning the bows and trying as quickly as possible to pass by the gazes of doctors, clergy and relatives fixed on him. Prince Vasily, who had lost weight and turned pale during these days, saw off the commander-in-chief and quietly repeated something to him several times. Having seen off the commander-in-chief, Prince Vasily sat down alone on a chair in the hall, crossing his legs high, resting his elbow on his knee and closing his eyes with his hand. After sitting like this for some time, he stood up and with unusually hasty steps, looking around with frightened eyes, walked through the long corridor to the back half of the house, to the eldest princess. Those in the dimly lit room spoke in an uneven whisper to each other and fell silent each time and, with eyes full of question and expectation, looked back at the door that led to the dying man’s chambers and made a faint sound when someone came out of it or entered it. “The human limit,” said the old man, a clergyman, to the lady who sat down next to him and naively listened to him, “the limit has been set, but you cannot pass it.” “I’m wondering if it’s too late to perform unction?” - adding the spiritual title, the lady asked, as if she had no opinion of her own on this matter.

The circumscribed circle of a quadrilateral. ? ? A circle can be described around a quadrilateral if the sum of the opposite angles is 180°: ? + ? =? + ? If a quadrilateral is inscribed in a circle, then the sum of the opposite angles is 180°. ? ? a. d. d1. PTOLOMY'S THEOREM The sum of the products of opposite sides is equal to the product of the diagonals: ac + bd = d1 d2. d2. b. c. b. Area of ​​a quadrilateral. a. c. d. Where p is the semi-perimeter of the quadrilateral.

Slide 9 from the presentation "Radius of inscribed and circumscribed circle". The size of the archive with the presentation is 716 KB.

Geometry 9th grade

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