1 circle definition arc of a circle central angle. Inscribed and circumscribed circles. Theorem on the product of segments of intersecting chords

Circle- a geometric figure consisting of all points of the plane located at a given distance from a given point.

This point (O) is called center of the circle.
Circle radius- this is a segment connecting the center with any point on the circle. All radii have the same length (by definition).
Chord- a segment connecting two points on a circle. A chord passing through the center of a circle is called diameter. The center of a circle is the midpoint of any diameter.
Any two points on a circle divide it into two parts. Each of these parts is called arc of a circle. The arc is called semicircle, if the segment connecting its ends is a diameter.
The length of a unit semicircle is denoted by π .
The sum of the degree measures of two arcs of a circle with common ends is equal to 360º.
The part of the plane bounded by a circle is called all around.
Circular sector- a part of a circle bounded by an arc and two radii connecting the ends of the arc to the center of the circle. The arc that limits the sector is called arc of the sector.
Two circles having a common center are called concentric.
Two circles intersecting at right angles are called orthogonal.

The relative position of a straight line and a circle

1. If the distance from the center of the circle to the straight line is less than the radius of the circle ( d), then the straight line and the circle have two common points. In this case the line is called secant in relation to the circle.
2. If the distance from the center of the circle to the straight line is equal to the radius of the circle, then the straight line and the circle have only one common point. This line is called tangent to the circle, and their common point is called point of tangency between a line and a circle.
3. If the distance from the center of the circle to the straight line is greater than the radius of the circle, then the straight line and the circle have no common points
.

Central and inscribed angles

Central angle is an angle with its vertex at the center of the circle.
Inscribed angle- an angle whose vertex lies on a circle and whose sides intersect the circle.

Inscribed angle theorem

An inscribed angle is measured by the half of the arc on which it subtends.

• Corollary 1.
  Inscribed angles subtending the same arc are equal.


• Corollary 2.
  An inscribed angle subtended by a semicircle is a right angle.

Theorem on the product of segments of intersecting chords.

If two chords of a circle intersect, then the product of the segments of one chord is equal to the product of the segments of the other chord.

Basic formulas

• Circumference:

• Circular arc length:

• Diameter:

• Circular arc length:

• Area of ​​a circle:

• Area of ​​the circular sector:

Equation of a circle

• In a rectangular coordinate system, the equation of a circle with radius is r centered at a point C(x o;y o) has the form:

• The equation of a circle of radius r with center at the origin has the form:

This article contains the minimum set of information about the circle required to successfully pass the Unified State Exam in mathematics.

Circumference is a set of points located at the same distance from a given point, which is called the center of the circle.

For any point lying on the circle, the equality is satisfied (The length of the segment is equal to the radius of the circle.

A line segment connecting two points on a circle is called chord.

A chord passing through the center of a circle is called diameter circle() .

Circumference:

Area of ​​a circle:

Arc of a circle:

The part of a circle enclosed between two points is called arc circles. Two points on a circle define two arcs. The chord subtends two arcs: and . Equal chords subtend equal arcs.

The angle between two radii is called central angle :

To find the arc length, we make a proportion:

a) the angle is given in degrees:

b) the angle is given in radians:

Diameter perpendicular to chord , divides this chord and the arcs that it subtends in half:

If chords And circles intersect at a point , then the products of the chord segments into which they are divided by a point are equal to each other:

Tangent to a circle.

A straight line that has one common point with a circle is called tangent to the circle. A straight line that has two points in common with a circle is called secant

A tangent to a circle is perpendicular to the radius drawn to the point of tangency.

If two tangents are drawn from a given point to a circle, then tangent segments are equal to each other and the center of the circle lies on the bisector of the angle with the vertex at this point:

If a tangent and a secant are drawn from a given point to a circle, then the square of the length of a tangent segment is equal to the product of the entire secant segment and its outer part :

Consequence: the product of the entire segment of one secant and its external part is equal to the product of the entire segment of another secant and its external part:

Angles in a circle.

The degree measure of the central angle is equal to the degree measure of the arc on which it rests:

An angle whose vertex lies on a circle and whose sides contain chords is called inscribed angle . An inscribed angle is measured by half the arc on which it rests:

∠∠

The inscribed angle subtended by the diameter is right:

∠∠∠

Inscribed angles subtended by one arc are equal :

Inscribed angles subtending one chord are equal or their sum is equal

∠∠

The vertices of triangles with a given base and equal vertex angles lie on the same circle:

Angle between two chords (an angle with a vertex inside a circle) is equal to half the sum of the angular values ​​of the arcs of a circle contained inside a given angle and inside a vertical angle.

∠ ∠∠(⌣ ⌣ )

Angle between two secants (an angle with a vertex outside the circle) is equal to the half-difference of the angular values ​​of the arcs of the circle contained inside the angle.

∠ ∠∠(⌣ ⌣ )

Inscribed circle.

The circle is called inscribed in a polygon , if it touches its sides. Center of inscribed circle lies at the intersection point of the bisectors of the angles of the polygon.

Not every polygon can fit a circle.

Area of ​​a polygon in which a circle is inscribed can be found using the formula

here is the semi-perimeter of the polygon, and is the radius of the inscribed circle.

From here inscribed circle radius equals

If a circle is inscribed in a convex quadrilateral, then the sums of the lengths of opposite sides are equal . Conversely: if in a convex quadrilateral the sums of the lengths of opposite sides are equal, then a circle can be inscribed in the quadrilateral:

You can inscribe a circle into any triangle, and only one. The center of the incircle lies at the point of intersection of the bisectors of the interior angles of the triangle.

Inscribed circle radius equal to . Here

Circumscribed circle.

The circle is called described about a polygon , if it passes through all the vertices of the polygon. The center of the circumcircle lies at the point of intersection of the perpendicular bisectors of the sides of the polygon. The radius is calculated as the radius of the circle circumscribed by the triangle defined by any three vertices of the given polygon:

A circle can be described around a quadrilateral if and only if the sum of its opposite angles is equal to .

Around any triangle you can describe a circle, and only one. Its center lies at the point of intersection of the perpendicular bisectors of the sides of the triangle:

Circumradius calculated using the formulas:

Where are the lengths of the sides of the triangle and is its area.

Ptolemy's theorem

In a cyclic quadrilateral, the product of the diagonals is equal to the sum of the products of its opposite sides:

Definition 2

A polygon that satisfies the condition of definition 1 is called circumscribed about a circle.

Figure 1. Inscribed circle

Theorem 1 (about a circle inscribed in a triangle)

Theorem 1

You can inscribe a circle into any triangle, and only one.

Proof.

Consider triangle $ABC$. Let's draw bisectors in it that intersect at point $O$ and draw perpendiculars from it to the sides of the triangle (Fig. 2)

Figure 2. Illustration of Theorem 1

Existence: Let us draw a circle with center at point $O$ and radius $OK.\ $Since point $O$ lies on three bisectors, it is equidistant from the sides of triangle $ABC$. That is, $OM=OK=OL$. Consequently, the constructed circle also passes through the points $M\ and\ L$. Since $OM,OK\ and\ OL$ are perpendiculars to the sides of the triangle, then by the circle tangent theorem, the constructed circle touches all three sides of the triangle. Therefore, due to the arbitrariness of a triangle, a circle can be inscribed in any triangle.

Uniqueness: Suppose that another circle with center at point $O"$ can be inscribed in triangle $ABC$. Its center is equidistant from the sides of the triangle, and, therefore, coincides with point $O$ and has a radius equal to length $OK$ But then this circle will coincide with the first one.

The theorem has been proven.

Corollary 1: The center of a circle inscribed in a triangle lies at the point of intersection of its bisectors.

Here are a few more facts related to the concept of an inscribed circle:

Not every quadrilateral can fit a circle.

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

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

Definition 3

If all the vertices of a polygon lie on a circle, then the circle is called circumscribed about the polygon (Fig. 3).

Definition 4

A polygon that satisfies definition 2 is said to be inscribed in a circle.

Figure 3. Circumscribed circle

Theorem 2 (about the circumcircle of a triangle)

Theorem 2

Around any triangle you can describe a circle, and only one.

Proof.

Consider triangle $ABC$. Let us draw perpendicular bisectors in it, intersecting at point $O$, and connect it with the vertices of the triangle (Fig. 4)

Figure 4. Illustration of Theorem 2

Existence: Let's construct a circle with center at point $O$ and radius $OC$. Point $O$ is equidistant from the vertices of the triangle, that is, $OA=OB=OC$. Consequently, the constructed circle passes through all the vertices of a given triangle, which means that it is circumscribed about this triangle.

Uniqueness: Suppose that another circle can be described around the triangle $ABC$ with its center at the point $O"$. Its center is equidistant from the vertices of the triangle, and, therefore, coincides with the point $O$ and has a radius equal to the length $OC. $ But then this circle will coincide with the first one.

The theorem has been proven.

Corollary 1: The center of the circle circumscribed about the triangle coincides with the point of intersection of its bisectoral perpendiculars.

Here are a few more facts related to the concept of a circumcircle:

It is not always possible to describe a circle around a quadrilateral.

In any cyclic quadrilateral, the sum of opposite angles is $(180)^0$.

If the sum of the opposite angles of a quadrilateral is $(180)^0$, then a circle can be drawn around it.

An example of a problem on the concepts of inscribed and circumscribed circles

Example 1

In an isosceles triangle, the base is 8 cm and the side is 5 cm. Find the radius of the inscribed circle.

Solution.

Consider triangle $ABC$. By Corollary 1, we know that the center of the incircle lies at the intersection of the bisectors. Let us draw the bisectors $AK$ and $BM$, which intersect at the point $O$. Let's draw a perpendicular $OH$ from point $O$ to side $BC$. Let's draw a picture:

Figure 5.

Since the triangle is isosceles, then $BM$ is both the median and the height. By the Pythagorean theorem $(BM)^2=(BC)^2-(MC)^2,\ BM=\sqrt((BC)^2-\frac((AC)^2)(4))=\sqrt (25-16)=\sqrt(9)=$3. $OM=OH=r$ -- the required radius of the inscribed circle. Since $MC$ and $CH$ are segments of intersecting tangents, then by the theorem on intersecting tangents, we have $CH=MC=4\cm$. Therefore, $BH=5-4=1\ cm$. $BO=3-r$. From the triangle $OHB$, according to the Pythagorean theorem, we obtain:

\[((3-r))^2=r^2+1\] \ \ \

Answer:$\frac(4)(3)$.

In this article we will analyze in great detail the definition of the number circle, find out its main property and arrange the numbers 1,2,3, etc. About how to mark other numbers on the circle (for example, \(\frac(π)(2), \frac(π)(3), \frac(7π)(4), 10π, -\frac(29π)( 6)\)) understands .

Number circle called a circle of unit radius whose points correspond , arranged according to the following rules:

1) The origin is at the extreme right point of the circle;

2) Counterclockwise - positive direction; clockwise – negative;

3) If we plot the distance \(t\) on the circle in the positive direction, then we will get to a point with the value \(t\);

4) If we plot the distance \(t\) on the circle in the negative direction, then we will get to a point with the value \(–t\).

Why is the circle called a number circle?
Because it has numbers on it. In this way, the circle is similar to the number axis - on the circle, like on the axis, there is a specific point for each number.

Why know what a number circle is?
Using the number circle, the values ​​of sines, cosines, tangents and cotangents are determined. Therefore, to know trigonometry and pass the Unified State Exam with 60+ points, you must understand what a number circle is and how to place dots on it.

What do the words “...of unit radius...” mean in the definition?
This means that the radius of this circle is equal to \(1\). And if we construct such a circle with the center at the origin, then it will intersect with the axes at points \(1\) and \(-1\).

It doesn’t have to be drawn small; you can change the “size” of the divisions along the axes, then the picture will be larger (see below).

Why is the radius exactly one? This is more convenient, because in this case, when calculating the circumference using the formula \(l=2πR\), we get:

The length of the number circle is \(2π\) or approximately \(6.28\).

What does “...the points of which correspond to real numbers” mean?
As we said above, on the number circle for any real number there will definitely be its “place” - a point that corresponds to this number.

Why determine the origin and direction on the number circle?
The main purpose of the number circle is to uniquely determine its point for each number. But how can you determine where to put the point if you don’t know where to count from and where to move?

Here it is important not to confuse the origin on the coordinate line and on the number circle - these are two different reference systems! And also do not confuse \(1\) on the \(x\) axis and \(0\) on the circle - these are points on different objects.

Which points correspond to the numbers \(1\), \(2\), etc.?

Remember, we assumed that the number circle has a radius of \(1\)? This will be our unit segment (by analogy with the number axis), which we will plot on the circle.

To mark a point on the number circle corresponding to the number 1, you need to go from 0 to a distance equal to the radius in the positive direction.

To mark a point on the circle corresponding to the number \(2\), you need to travel a distance equal to two radii from the origin, so that \(3\) is a distance equal to three radii, etc.

When looking at this picture, you may have 2 questions:
1. What happens when the circle “ends” (i.e. we make a full revolution)?
Answer: let's go for the second round! And when the second one is over, we’ll go to the third one and so on. Therefore, an infinite number of numbers can be plotted on a circle.

2. Where will the negative numbers be?
Answer: right there! They can also be arranged, counting from zero the required number of radii, but now in a negative direction.

Unfortunately, it is difficult to denote integers on the number circle. This is due to the fact that the length of the number circle will not be equal to an integer: \(2π\). And at the most convenient places (at the points of intersection with the axes) there will also be fractions, not integers

Lecture: Circle and Circle

Circle is a closed curve, all points of which are at the same distance from the center.

In everyday life, you have seen a circle more than once. This is exactly what the hour and second hand describes, and it is the shape of a circle that a gymnastics hoop has.

Now imagine that you drew a circle on a piece of paper and wanted to color it.

So, all the decorated space, limited by a circle, is a circle.

The center is the point that is equidistant from all points on the circle. The center of a circle and circle is designated by the letter O.

Radius is the distance from the center to the circle (R).

Diameter is a line segment passing through the center that connects all points of the circle (d). Moreover, the diameter is equal to two radii: d = 2R.

A chord is a segment that connects any two points on a circle. Diameter is a special case of chord.

To find the circumference, you need to use the formula:

l=2 πR

Please note that the circumference and area depend only on the radius of the circle.

The area of ​​a circle can be found using the following formula:

S=πR 2 .

I would like to draw your attention to the number “Pi”. This value was found using a circle. To do this, its length was divided into two radii, and thus the number “Pi” was obtained.

If a circle is divided into some parts with two radii, then such parts will be called sectors. Each sector has its own degree measure - the degree measure of the arc on which it rests.

To find the arc length, you need to use the formula:

2. Using radian measure:

If the vertex of a certain angle rests on the center of the circle, and its rays intersect the circle, then such an angle is called central.

If some two chords intersect at some point, then their segments are proportional: