Presentation on the theme of the circle in kindergarten. Presentation "circle and circle" presentation for a geometry lesson on the topic. What is the radius of a circle

CIRCLE AND CIRCLE

MATHEMATICS - 5 cells

Goals and objectives of the lesson:

Tutorials:

Ensure the assimilation of the concepts of a circle, a circle and their elements (radius, diameter, chord, arc).
Consider the relationship between diameter and radius of a circle.
To introduce the compass tool, to teach how to draw a circle with a compass.
Learn to find common and different between a circle and a circle; broaden the horizons of students.

Developing:

The development of logical thinking, attention, creative and cognitive abilities, imagination, the ability to analyze, draw conclusions.
Formation of accuracy and accuracy in the execution of drawings.
The use of information technology in the study of mathematics.

Educational:

Development of diligence, discipline, respect for classmates.
Formation of interest in mathematics.

Equipment: interactive whiteboard, computer, drawing tools.

The compass is a drawing tool. It has a needle on one end and a pencil on the other.

The circle must be handled with care!

1. Mark a point in your notebook and mark it with the letter O.

2. Take a compass, spread the "legs" of the compass to a distance of 3 cm.

3. Place the needle of the compass at point O, and draw a closed line with the other “leg” of the compass.

We got a closed line, which is called circle . What is a circle?

Task number 1: Which figure shows a circle and why.

Circle – a geometric figure consisting of all points located at the same distance from a given point. This point is called circle center .

Circle - This is the simplest of the curved lines. One of the oldest geometric figures. Aristotle argued that the planets and stars should move along the most perfect line - the circle. For hundreds of years, astronomers believed that the planets move in a circle. Only in the 17th century, scientists: Copernicus, Galileo, Kepler, Newton refuted this opinion.

Task 2

1) Draw a circle centered at O.

2) On the circle mark three points A, B and C.

3) Connect them with a segment to the center of the circle.

4) What can be said about the resulting segments?

Conclusion: All segments are equal, because All points on a circle are the same distance from the center.

This distance is called the radius, denoted by - r .

What is the radius of a circle?

Circle radius is a line segment that connects the center of the circle and a point on the circle.

Even the Babylonians and the ancient Indians considered the most important element of the circle - radius. The word is mathematical and means "beam".

In ancient times, this term did not exist. Euclid and other scientists simply said "straight from the center", then in the 11th century it was called "half-diameter". The term "radius" is first encountered in 1569 by the French scientist Rams. Generally accepted - "radius" becomes only in the 17th century.

Euclid -

Great Ancient Greek

mathematician; first

mathematician of Alexandria

schools

Construct two circles in a notebook with a radius of 2 cm. Paint over the inner area of one circle.

A circle

Circle

How are the two drawings similar and how are they different?

A CIRCLE - a geometric figure consisting of all points of the plane that are inside the circle (including the circle itself).

CIRCLE - a geometric figure consisting of all points located at the same distance from the center of the circle.

Which objects are circle-shaped and which ones are circle-shaped?

Task 3

Construct a circle centered at point O, r = 3 cm. Mark two points A and B on the circle and connect them with a segment.

AB - chord

Chord A line segment that connects two points on a circle.

Chord - this Greek word "chorde" - a string, was introduced by European scientists in the 12-13th centuries. A chord divides a circle into two arcs.

CD = r+r = 2r = d = 2r "width="640"

Task 4

Draw a chord through the center of the circle.

This chord is called - diameter, denoted – d.

Define diameter.

Circle diameter is a chord passing through the center of the circle.

CD = OC+OD, OC = r, OD = r = CD = r+r = 2r = d = 2r

The diameter is made up of two radii, so the diameter is twice as long as the radius. The radius is twice the diameter.
So, diameter is 2 radii, and then the radius is half the diameter. r = 4 cm, d=2 r, d = 2 4 = 8 cm d = 8 cm, r=d:2, r = 8:2 = 4 cm
Memorize these formulas!

d=2 r

How are radius and diameter related?

Extend line segment AO until it intersects the circle.

Mark the point of intersection with the letter K.

The segment AK is called diameter circles.

Diameter denoted by the Latin letter d.

Circle diameter is a line segment that connects two points on a circle and passes through its center.

connect the dots

M and K, A and M.

The segments MK and AM are called chords circles.

Chord is a line segment that connects two points on a circle.

Name all the radii, diameters and chords of a circle.

Draw a circle centered at point O.

Mark two points A and B on the circle.

Points A and B divide the circle into two parts, which are called arcs circles.

Formulate definition of arc circles.

arc of a circle is the part of a circle enclosed between two of its points.

Name all arcs on a circle:

dots,

lying on a circle.

dots,

not lying on a circle.

dots,

lying on a circle.

Test

Option 2

A1. What is the name of the segment AB in drawing No. 2?

1) chord of a circle

2) circle diameter

3) circle radius

A2. Choose the correct sentence of the statement:

The diameter of a circle is the line segment that...

A3. Can a circle have two radii of different lengths?

2) can't

3) find it difficult to answer

Option 1

A1. What is the name of the segment AB in drawing No. 1?

1) circle diameter

2) circle radius

3) chord of a circle

A2. Choose the correct continuation of the statement:

The radius of a circle is a line segment that...

1) connects any two points of the circle

2) connects the center of the circle to any point on the circle

3) connects two points of the circle and passes through the center of the circle

A3. Can a circle have two diameters of different lengths?

2) can't

3) make it difficult to answer

test yourself

Draw a circle with a center at point O and a radius of 3 cm. Draw a straight line that intersects the circle at points M and K.

How far from the center of the circle are these points?

The segments OM and OK are the radii of the circle, therefore

OM=3cm, OK=3cm

Solution

Answer: at a distance of 3 cm

Task number 1

A segment AB is given, its length is 4 cm. Construct a point X if it is known that AX = 3 cm, BX = 5 cm.

How many points did you get?

Solution

Answer: two dots

Task number 2

The segment AB is the same as in the previous task, its length is 4 cm. Construct a point X if it is known that: 1) AX = 1 cm, BX = 3 cm. 2) AX = 1 cm, BX = 2 cm. points you received in the first case and how many in the second case?

Solution

Answer: none!

Answer: one dot

Task number 3

The radius of the circle with center O is 2 cm. Position points A, B, C so that: the distance from O to A is less than 2 cm, the distance from O to B is 2 cm, the distance from C to O is more than 2 cm.

Solution

2 cm

Answer: point A can be located anywhere inside the circle; point B - on the circle; point C - anywhere outside the circle

Summary of the lesson (reflection):

Describe your impressions about today's lesson:

I found out…
I can…
It was difficult…
I like it…
Thanks for…

Homework

pp. 133-134, memo (learn definitions),
Ex. 855, 874, 875, 876.
Extra . Make a pattern of circles (ornament).

Thanks to all for work!

Math lesson in 5th grade

on the topic "Circumference and Circle".

©GBOU boarding school №1
Mathematics teacher: Makarova N.A.
St. Petersburg, 2015.

Goals and objectives of the lesson:

Tutorials:

Ensure the assimilation of the concepts of a circle, a circle and their elements (radius, diameter, chord, arc).
Consider the relationship between diameter and radius of a circle.
To introduce the compass tool, to teach how to draw a circle with a compass.
Learn to find common and different between a circle and a circle; broaden the horizons of students.

Developing:

The development of logical thinking, attention, creative and cognitive abilities, imagination, the ability to analyze, draw conclusions.
Formation of accuracy and accuracy in the execution of drawings.
The use of information technology in the study of mathematics.

Educational:

Development of diligence, discipline, respect for classmates.
Formation of interest in mathematics.

Equipment: interactive whiteboard, computer, drawing tools.

The compass is a drawing tool. It has a needle on one end and a pencil on the other.

The circle must be handled with care!

1. Mark a point in your notebook and name it with the letter O.

2. Take a compass, spread the "legs" of the compass to a distance of 3 cm.

3. Place the needle of the compass at point O, and draw a closed line with the other “leg” of the compass.

A circle is a closed line consisting of points that are equally distant from the center.

Point O is called the center of the circle.

Mark two points A and M on the circle.

The segments OA and OM are called the radii of the circle.

The radius of a circle is a line segment that connects the center of the circle and a point on the circle.

Connect the points O and M, O and A.

The radius is denoted

latin letter r.

Construct two circles in a notebook with a radius of 2 cm. Paint over the inner area of one circle.

CIRCLE - a geometric figure consisting of all points located at the same distance from the center of the circle.

CIRCLE - a geometric figure consisting of all points of the plane that are inside the circle (including the circle itself).

Circle

Which objects are circle-shaped and which ones are circle-shaped?

Extend line segment AO until it intersects the circle.

Mark the point of intersection with the letter K.

The segment AK is called the diameter of the circle.

The diameter of a circle is a line segment that connects two points on a circle and passes through its center.

The diameter is indicated by the Latin letter d.

connect the dots

M and K, A and M.

Segments MK and AM are called chords of a circle.

A chord is a line segment that connects two points on a circle.

Name all the radii, diameters and chords of a circle.

Draw a circle centered at point O.

Mark two points A and B on the circle.

Points A and B divide the circle into two parts, which are called arcs of the circle.

An arc of a circle is a part of a circle

between points A and B.

Name all arcs on a circle:

Name the points

lying on a circle.

Name the points

not lying on a circle.

Name the points

lying on a circle.

Option 1

A1. What is the name of the segment AB in drawing No. 1?

1) circle diameter

2) circle radius

3) chord of a circle

A2. Choose the correct continuation of the statement:

The radius of a circle is a line segment that...

A3. Can a circle have two diameters of different lengths?

2) can't

3) make it difficult to answer

Option 2

A1. What is the name of the segment AB in drawing No. 2?

1) chord of a circle

2) circle diameter

3) circle radius

A2. Choose the correct sentence of the statement:

The diameter of a circle is the line segment that...

1) connects any two points of the circle

2) connects the center of the circle to any point on the circle

3) connects two points of the circle and passes through the center of the circle

A3. Can a circle have two radii of different lengths?

2) can't

3) find it difficult to answer

Back forward

Attention! The slide preview is for informational purposes only and may not represent the full extent of the presentation. If you are interested in this work, please download the full version.

The first lesson in the topic "Ordinary fractions".

Textbook by N.Ya. Vilenkin “Mathematics 5”.

The objectives of the lesson: to familiarize students with the concept of a circle and a circle; the formation of the ability to build a circle using a compass for a given radius and diameter.

Learning objectives aimed at achieving:

Personal development:

continue to develop the ability to clearly, accurately and competently express their thoughts in oral and written speech,
develop creative thinking, initiative, resourcefulness, activity in solving mathematical problems.

Meta-subject development:

broaden horizons, instill the ability to work together (a sense of camaraderie and responsibility for the results of their work);
continue to develop the ability to understand and use mathematical visual aids.

Subject development:

to form a theoretical and practical idea of a circle and a circle, as about geometric figures, their elements;
continue the development of visual skills (learn to use a compass to build a circle of any radius);
to form the ability to apply the learned concepts to solve practical problems.

Type of lesson: a lesson in obtaining new knowledge, skills and abilities.

Forms of student work:

individual;
frontal;
independent work;
work in pairs;
test control.

Necessary equipment:

Projector and screen.
Presentation "Circle and circle".
Individual sheet for each student Attachment 1).

Structure and course of the lesson

	Lesson stage	slide number	Teacher activity	Student activities	Formation of UUD (personal, metasubject)	Time (in minutes)
1.	Organizing time	№1,2	welcomes students, sets them up for work, offers to check the readiness of the workplace, poses problems, using a poem designed in the presentation.	greet teachers, check readiness for the lesson, They express their opinion on the question posed by comparing the figures: a circle and a circle.	cognitive (the ability to solve educational problems that arise in the course of frontal work).	2
2	Knowledge update. Formulation of the problem.	№3	declares the objectives of the lesson, writes down the date and the topic of the lesson - “Circumference and Circle”.	write down the date and topic of the lesson in a notebook.	Regulatory (capacity for volitional effort)	1
3.	“Discovery” of new knowledge by children.	№4	Conducts a frontal survey according to the drawing on the slide. 1. Which of the drawn figures can be called lines?	Answer the teacher's questions and write down the answers in individual sheets.	cognitive (the ability to read meaningfully, extracting the necessary information; the ability to search and extract the necessary information)	5
			2. Which of them are broken lines, which ones are curves?	2. №2,4
			3. Divide curved lines into closed and open.	3. Closed - 3.6.8 open -1.5.9
			4. Points are placed in closed curves 3,6,8, can it be argued that the distance from point O to points A, B, C, D is the same in each figure? Measure the distance to these points using a ruler. Write down the answers.	4. Students measure the distance from point O to points A, B, C, D. Record the results on individual sheets.
			5. Compare figures 6 and 8.	5. Similarity: these are closed curved lines, point O is marked inside, and points A, B, C, D are marked on the lines. Difference: distance from point O to points A, B, C, D in figure 6 - different, in figure 8 - the same
			6. Why do you think figure 8 is a circle, and figure 6 is not a circle?	6. Because in figure 8 the distance from point O to points A, B, C, D are the same, and in figure 6 they are different
			7. What are the essential features of a circle!	7. This is a curved closed line; the distance from point O to all points on the circle is the same.
			8. Can figures 5, 7,9 be called circles?	8. NO! Figures 9 and 5 are not closed curves, and figure 7 does not have a center, the distances from which to all points on the circle are the same.
			9. What is the difference between circles 3 and 8?	9. The distance from point O to points on the circle!
			10. Mark any other point on circle 8 and measure the distance from point O - the center of the circle - to this point, draw a conclusion!	10. The distance from the center of the circle to any point on the circle is the same!
4		№5,6	Preparing students for the next stage of the lesson. Riddle about the compass in verse. Safety precautions for working with a compass. With the help of slides, the presentation clearly shows the structure of the compass and its purpose.	Guess the riddle - "Compass" Find all the elements on your compass.	Communicative (ability to engage in dialogue)	2
5.	The study of new material and its primary consolidation.	№7,8	The teacher invites students to build a circle of arbitrary radius with him.	Do the task of the teacher.	cognitive(the ability to make a model and transform it if necessary). Communication (ability to hear and listen) Regulatory(ability to analyze the course and method of action)	15
		№9	Offers to remember which familiar objects have the shape of a circle, and which are the shape of a circle?	List items
		№10, 11	Introduces new concepts “circle center”, “circle radius”
		№12	Offers students, without violating the patterns, to build radii in the last circles in the research sheet. Then it includes correctly constructed radii on the slide.	Build radii and explain what pattern they have identified. Check for correctness.
		№13	Invites students to do independent research: Build a circle with a radius of 3 cm and mark its center. Connect two points of the circle so that this segment passes through the center of the circle. Defines the "diameter of a circle".	They complete the task in individual sheets, draw a conclusion, then check and correct their mistakes using the presentation slides.
		№14	Write an expression to find the length of this segment. Then he asks the students to check their research using the presentation slide.	Students make appropriate entries in their notebooks.
		№15	Introduces the concept of “circle chord”.	Students make appropriate entries in their notebooks.
		№16	Gives the task to students: list all the diameters, chords and radii of a circle.
		№17	Introduction of a new concept “arc of a circle”.	Students make appropriate entries in their notebooks.
		№18	Gives a task: name all the arcs of a circle.	Orally perform the task of the teacher.
		№19	He proposes to perform a practical task: using a compass, build two circles in a notebook with the same radius equal to 3 cm, paint over the inner area of one circle. He asks the question: how can one explain that the first figure is called a circle, and not a circle?	Perform the construction of figures in an individual sheet, and call the resulting figures. They answer the question: The first figure is painted over, i.e. all points inside this figure belong to it, and it is called a circle.
		№20	Task: name the points lying in the inner (outer) area.	Orally perform the task of the teacher.
6.	Research work in pairs.	№21	Gives assignments and provides advisory assistance to students who have difficulties.	Perform work in pairs.	Communicative (the ability to cooperate with other people in finding the necessary information)	10
7.	Test work with mutual control.	№22	Invites students to test their knowledge with a quiz.	Students complete the test, followed by mutual control.		2
8.	Summary of the lesson.	№23	Summarizes the lesson. He offers to describe his impressions of today's lesson and draw a smile to the emoticon, depending on the mood of the students. Sets homework:	Describe in individual sheets the impressions of the research activities carried out, their impressions and their emotional state. Record homework in a diary.		3

TEST Find: sector, arc, radius, diameter, chord, segment

Through three points A, B and C that do not lie on one straight line (through the vertices of ABC), it is possible to draw a circle if there is such a fourth point. O, which is equally distant from points A, B, and C. Let us prove that such a point exists and, moreover, only one. Any point equidistant from points A and B must lie on the perpendicular bisector MN to segment AB, just like any point equidistant from points B and C must lie on the perpendicular bisector PQ drawn to the side BC. Hence, if there exists a point equidistant from the three points A, B, and C, then it must lie on both MN and PQ, which is possible only if it coincides with the point of intersection of these two lines. The lines MN and PQ always intersect because they are perpendicular to the intersecting lines AB and BC. The point O of their intersection will be a point equally distant from A, from B and from C, which means that if we take this point as the center, and take the distance OA (or OB, or OC) as the radius, then the circle will pass through points A, B and C. Since the lines MN and PQ can intersect at only one point, there can be only one center of the circle and the length of its radius can be only one; hence the desired circle is unique.

Let's bend the drawing along the diameter AB so that its left side falls on the right. Then the left semicircle will coincide with the right semicircle and the perpendicular CS will go along KD. From this it follows that the point C, which is the intersection of the semicircle with the CS, will fall on D; therefore CK= KD; BC=BD, AC=AD. BC= BD AC= AD

Properties of the diameter of a circle 1. The diameter drawn through the middle of a chord is perpendicular to this chord and divides the arc subtracted by it in half. 2. The diameter drawn through the middle of the arc is perpendicular to the chord subtending this arc and divides it in half.

1. Consider a circle with center O. AB \u003d CD, P is the middle of the chord AB, Q is the middle of CD. 2. Consider ΔОАР and ΔOCQ (rectangular): ОА = OS - radii, PA = CQ - half equal chords 3.ΔОАР = ΔOCQ (along the hypotenuse and leg). From the equality of triangles OP = OQ (equal legs), i.e. chords are equidistant from the center

Cases of mutual arrangement of a straight line and a circle d rd > r rd > r"> rd > r"> rd > r" title="(!LANG: Cases of mutual position of a line and a circle d rd > r"> title="Cases of mutual arrangement of a straight line and a circle d rd > r"> !}

D>r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle do not have common points. O d>r r r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle have no common points. O d>rr"> r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle do not have common points. O d>rr"> r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle do not have common points. O d>r r" title="(!LANG:d>r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle have no common points. O d>r r"> title="d>r If the distance from the center of the circle to the line is greater than the radius of the circle, then the line and the circle have no common points. O d>r r"> !}

Tangent property. Let the line p touch the circle at point A, that is, A is their only common point. Proof "by contradiction": 1. Let's assume that p is not perpendicular to the radius OA. Let's draw a perpendicular OB on the river. 2. Set aside on p the segment BC = BA. 3. OVA \u003d OBC (on two legs). Therefore OS = OA. 4. C lies on the circle. Therefore, p and the circle have two points in common, which is impossible. So, p OA, as required

Take any point A of the circle F and draw the radius OA. Then draw a line p perpendicular to the radius OA. Any point B of the straight line p, different from point A, is removed from O by more than a radius, since the inclined OB is longer than the perpendicular OA. Therefore, the point B does not lie on F. Hence, the point A is the only common point of p and F, i.e., p touches F at the point A.

Various cases of relative positions of two circles. d>R+R 1d>R+R 1 d=R+R 1d=R+R 1 d R+R 1d>R+R 1 d=R+R 1d=R+R 1 d"> R+R 1d>R+R 1 d=R+R 1d=R+R 1 d"> R+R 1d >R+R 1 d=R+R 1d=R+R 1 d" title="(!LANG: Different cases of relative position of two circles. d>R+R 1d>R+R 1 d=R+R 1d= R+R 1 d"> title="Various cases of relative positions of two circles. d>R+R 1d>R+R 1 d=R+R 1d=R+R 1 d"> !}

1. The circles lie one outside the other, without touching in this case, obviously, d\u003e R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles "> R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles"> R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles" title="(!LANG:1. The circles lie one outside the other, without touching in this case, obviously, d > R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles"> title="1. The circles lie one outside the other, without touching in this case, obviously, d\u003e R + R 1 R and R 1 - the radii of the circles d - the distance between the centers of the circles"> !}

3. Circles intersect then d

5. One circle lies inside the other without touching, then, obviously, d

R + R 1, then the circles are located one outside the other, without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R – R 1, then the circles intersect. 4. If d \u003d R - R 1, then the circles touch from the inside. 5." title="(!LANG: Inverse sentences 1. If d > R + R 1, then the circles are located one outside the other without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R - R 1, then the circles intersect. 4. If d = R - R 1, then the circles touch from the inside. 5." class="link_thumb"> 59 !} Reverse propositions 1. If d > R + R 1, then the circles are located one outside the other without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R – R 1, then the circles intersect. 4. If d \u003d R - R 1, then the circles touch from the inside. 5. If d R + R 1, then the circles are located one outside the other without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R – R 1, then the circles intersect. 4. If d \u003d R - R 1, then the circles touch from the inside. 5."> R + R 1, then the circles are located one outside the other, not touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R - R 1, then the circles intersect. 4. If d = R – R 1, then the circles touch from the inside 5. If d R + R 1, then the circles are located one outside the other without touching 2. If d = R + R 1, then the circles touch from the outside 3. If d R - R 1, then the circles intersect. 4. If d = R - R 1, then the circles touch from the inside. 5. " title="(!LANG:Reverse sentences 1. If d > R + R 1, then the circles are located one outside the other without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R – R 1, then the circles intersect. 4. If d = R – R 1, then the circles touch from the inside. 5."> title="Reverse propositions 1. If d > R + R 1, then the circles are located one outside the other without touching. 2. If d = R + R 1, then the circles touch from the outside. 3. If d R – R 1, then the circles intersect. 4. If d \u003d R - R 1, then the circles touch from the inside. five.">!}

Given: a circle with center O, ABC - inscribed Prove: ABC = ½ AC Proof: Consider the case when the side BC passes through the center O 1. Arc AC is less than a semicircle, AOC = AC (central) 2. Consider radii). ΔABO isosceles 1 = 2, AOC is the external angle ΔABO, AOC = = 2 1, therefore ABC = ½ AC 1 2

Given: circle with center O, ABC - inscribed Prove: ABC = ½ AC Proof: Consider the case when the center O lies inside the inscribed angle. 1. Additional construction: diameter BD 2. Beam BO divides ABC into two angles 3. Beam BO intersects arc AC at point D 4. AC = AD + DC, therefore ABD = ½ AD and DBC = ½ DC or ABD + DBC = ½ AD + ½ DC or ABC = ½ AC

Given: circle with center O, ABC - inscribed Prove: ABC = ½ AC Proof: Consider the case when the center O lies outside the inscribed angle. 1. Additional construction: diameter BD 2. Beam BO does not divide ABC into two angles 3. Beam BO does not intersect arc AC at point D 4. AC = AD - CD, therefore ABD = ½ AD and DBC = ½ DC or ABD - DBC = ½ AD - ½ DC or ABC = ½ AC

Proof. 1. Consider an arbitrary triangle ABC. We denote by the letter O the point of intersection of the medial perpendiculars to its sides and draw the segments O A, O B and OS. 2. Since the point O is equidistant from the vertices of the triangle ABC, then OA \u003d OB \u003d OS. Therefore, the circle with the center O of radius O A passes through all three vertices of the triangle and, therefore, is circumscribed about the triangle ABC. Proof. 1. Consider an arbitrary triangle ABC and denote by the letter O the point of intersection of its bisectors. 2. Let's draw perpendiculars OK from the point O. OL and OM, respectively, to the sides AB, BC and CA. 3. Since the point O is equidistant from the sides of the triangle ABC, then OK \u003d OL \u003d OM. Therefore, the circle with center O of radius OK passes through the points K, L and M. 4. The sides of the triangle ABC touch this circle at the points K, L, M, since they are perpendicular to the radii OK, OL and OM. Hence, the circle with center O of radius OK is inscribed in triangle ABC.

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