(a geometric figure)

In a plane, a closed curve formed by a moving point rotating around a certain point with a certain length is called a circle.

On the same plane, the set of points whose distance to a fixed point is equal to a fixed length is called a circle. A circle can be expressed as a set {m |||| mo | = r}, and the standard equation of a circle is (x-a)? + (y - b)？ = r？ . Where (a, b) is the center of the circle and r is the radius.

A circle is a conic curve, which is obtained by cutting a cone from a plane parallel to the bottom of the cone.

A circle is a geometric figure. By definition, compasses are usually used to draw circles. The diameter and radius length of the inner circle of the same circle are always the same, and the circle has countless radii and diameters. A circle is a figure with axial symmetry and central symmetry. The axis of symmetry is the straight line where the diameter lies. At the same time, the circle is a "positive infinite polygon" and "infinity" is just a concept. The more sides a polygon has, the closer its shape, perimeter and area are to a circle. So there is no real circle in the world, and the circle is actually just a conceptual figure.

path

1. The line segment connecting the center of the circle and any point on the circle is called radius, and the letter is expressed as r (radius).

2. The line segment whose two ends pass through the center of the circle is called diameter, and the letter is expressed as D (diameter). A straight line with a diameter is the symmetry axis of a circle.

Diameter of circle d=2r

bowstring

1. A line segment connecting any two points on a circle is called a chord. The longest chord in the same circle is the diameter. The straight line with the diameter is the symmetry axis of the circle, so there are countless symmetry axes of the circle.

arc

1. The part between any two points on the circle is called arc, and the abbreviation of arc is "⌒";

2. The arc larger than the semicircle is called the optimal arc, and the arc smaller than the semicircle is called the suboptimal arc, so the semicircle is neither the optimal arc nor the suboptimal arc. The optimal arc is generally represented by three letters, and the suboptimal arc is generally represented by two letters. The optimal arc is an arc with a central angle greater than 180 degrees, and the suboptimal arc is an arc with a central angle less than 180 degrees.

3. In the same circle or equal circle, two arcs that can overlap each other are called equal arcs.

corner

1. The angle of the vertex on the center of the circle is called the central angle.

2. The angle where the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle. The angle of a circle is equal to half the central angle of the same arc.

circumference ratio

The ratio of the circumference of a circle to the length of its diameter is called pi. It is an infinite cyclic decimal, usually expressed by letters.

Say,

≈3. 14 15926535 ... The approximate value is usually 3. 14. We can say that the circumference of a circle is π times the diameter, or about 3. 14 times the diameter, but we can't directly say that the circumference of a circle is 3. 14 times the diameter.

shape

1. The figure surrounded by the chord and the arc it faces is called an arch.

2. The figure enclosed by two radii of the central angle and an arc corresponding to the central angle is called a sector.

expression

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Circle-⊙; Radius —r or r (the letter represented by the radius of the outer ring in the ring); Center of the circle-o; Arc-⌒; Diameter —d? ;

Sector arc length -l? ; Circumference -c? ; Area -s

computing formula

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Circumference formula of circle

Circumference of the circle:

Half of the circumference c=πr

The circumference of a semicircle c=πr+2r.

Derivation of the formula of the circumference of a circle (this aspect involves arc differentiation)

Let the parameter equation of the circle be

Integral of a circle in a week

Substitute, available

that is

Area formula of circle

Formula for calculating circle area:

or

Find the diameter from the area of the circle:

Divide a circle into several equal parts and you can make an approximate rectangle. The width of a rectangle corresponds to the radius of a circle.

Cone lateral area

(l is the bus length)

Arc length angle formula

Sector arc length L= central angle (radian system) * R= nπR/ 180(θ is central angle) (r is sector radius).

Sector area S=nπ R? /360=LR/2(L is the arc length of the sector)

Radius of cone bottom surface r=nR/360(r is the radius of bottom surface) (N is the central angle)

Sector area formula

R is the radius of the sector, n is the degree of the central angle of the arc, π is pi, and L is the arc length corresponding to the sector.

You can also divide the area of the circle where the sector is located by 360 and multiply it by the angle n of the central angle of the sector, as shown below:

(l is arc length and r is sector radius)

Deduction process: S=πr? ×L/2πr=LR/2

(L=│α│ R)

location

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Position relationship between point and circle

①P is outside the circle of O, then PO>r.

②P is on the circle o, then po = r.

③P is in the circle o, then po.

or vice versa, Dallas to the auditorium

What are the points P(x0, y0) and the circle (x-a) in the plane? +(y-b)？ =r？ The general method to judge the positional relationship is:

① What if (x0-a)? +(y0-b)？ & ltr？ Then p is in the circle.

② What if (x0-a)? +(y0-b)？ =r？ , then p is on the circle.

③ What if (x0-a)? +(y0-b)？ & gtr？ , then p is outside the circle.

The positional relationship between straight line and circle

A straight line and a circle have nothing in common, which is called separation. AB is separated from circle O, d>r.

② A straight line and a circle have two common points, which are called intersections. This straight line is called the secant of a circle. AB intersects with o and d

③ A straight line and a circle have only one common point, which is called tangency. This straight line is called the tangent of the circle, and the only common point is called the tangent point. The line between the center of the circle and the tangent point is perpendicular to the tangent line. AB is tangent to ⊙O, and d = r. (d is the distance from the center of the circle to the straight line)

Line Ax+By+C=0 and circle x in the plane? +y？ The general method for judging the positional relationship of +Dx+Ey+F=0 is:

1. from Ax+By+C=0, you can get y=(-C-Ax)/B (where b is not equal to 0) and substitute it into x? +y？ +Dx+Ey+F=0, it becomes an equation about x.

If B2-4ac >: 0, a circle and a straight line have two common points, that is, the circle and the straight line intersect.

If b2-4ac=0, the circle and the straight line have 1 points in common, that is, the circle is tangent to the straight line.

If B2-4ac

2. if B=0, that is, the straight line is Ax+C=0, that is, x=-C/A, parallel to the y axis (or perpendicular to the x axis), put x? +y？ +Dx+Ey+F=0 decimal (x-a)? +(y-b)？ =r？ Let y=b, then find two x values x 1 and x2, and specify X 1

When x =-c/a

When x 1

Relationship between circle and its position

(1) have nothing in common. A circle is called the outside outside of another circle, and the inside is called the inside.

(2) If there is only one common point, a circle is said to be circumscribed by another circle and inscribed by another circle.

(3) There are two common points called intersection. The distance between the centers of two circles is called the center distance.

Let the radii of two circles be r and r respectively, and r > r, and the center distance is p, then the conclusion is: the outer distance is p>R+r; Circumscribed p = r+r; Including p<r-r;

Inner cut p = r-r; Intersecting r-r

Nature of circle

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The (1) circle is an axisymmetric figure, and its symmetry axis is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.

Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

Inverse theorem of vertical diameter theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting two arcs opposite to the chord.

⑵ The properties and theorems of central angle and central angle.

(1) In the same circle or the same circle, if one of two central angles, two peripheral angles, two sets of arcs, two chords and the distance between two chords is equal, their corresponding other groups are equal respectively.

(2) In the same circle or equal circle, the circumferential angle of an equal arc is equal to half of the central angle it faces (the circumferential angle and the central angle are on the same side of the chord).

The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.

The formula for calculating the central angle is θ = (l/2π r) × 360 =180l/π r = l/r (radian).

That is, the degree of the central angle is equal to the degree of the arc it faces; The angle of a circle is equal to half the angle of the arc it faces.

(3) If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc.

⑶ Properties and theorems about circumscribed circle and inscribed circle.

① A triangle has a unique circumscribed circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal;

(2) The center of the inscribed circle is the intersection of the bisectors of the inner angles of the triangle, and the distances to the three sides of the triangle are equal.

③R=2S△÷L(R: radius of inscribed circle, s: area of triangle, l: perimeter of triangle).

(4) The intersection of the connecting lines of two tangent circles. (line: a straight line with two centers connected)

⑤ The midpoint M of the chord PQ on the circle O, if the intersection point M is two chords AB and CD, and the chords AC and BD intersect PQ on X and Y respectively, then M is the midpoint of XY.

(4) If two circles intersect, the line segment (or straight line) connecting the centers of the two circles bisects the common chord vertically.

(5) The degree of the chord tangent angle is equal to half the degree of the arc it encloses.

(6) The degree of the angle inside a circle is equal to half of the sum of the degrees of the arcs subtended by the angle.

(7) The degree of the outer angle of a circle is equal to half of the difference between the degrees of two arcs cut by this angle.

(8) The perimeters are equal, and the area of a circle is larger than that of a square, rectangle or triangle.

Correlation theorem

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Tangent theorem

The radius perpendicular to the tangent point; The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

Judgment method of tangent: the straight line passing through the outer end of radius and perpendicular to this radius is the tangent of the circle.

Properties of tangent:

(1) The straight line passing through the tangent point and perpendicular to the radius of the tangent point is the tangent of the circle.

(2) The straight line perpendicular to the tangent point must pass through the center of the circle.

(3) The tangent of the circle is perpendicular to the radius passing through the tangent point.

Tangent length theorem

Length of two tangents from a point outside the circle to the circle, etc. , the line between this point and the center of the circle bisects the included angle of the tangent.

Here is a brief introduction to the proof of tangent length theorem.

Want to prove AC? =? AB, it proves that △ ABO △ ACO.

Let OC and OB be two radii of a circle, and ∠ABO? = ∠ACO=90

In Rt△ABO and Rt△ACO.

∴Rt△ABO？ ≌ Rt△ACO(H.L)

∴AB=AC, and ∞∠AOB =∠AOC, and ∞∠OAB =∠oac.

Tangent secant theorem

Proof of cutting line theorem;

The tangent and secant of the circle intersect at point P, the tangent intersects at point C, and the secant intersects at points A and B, so PC 2 = PA Pb.

Let ABP be the secant of ⊙O, PT be the tangent of ⊙O, and the tangent point is T, then PT? =PA PB

Proof: connected to, BT

∫∠PTB =∣∠Pat (tangent angle theorem)

∠APT=∠TPB (public corner)

∴△PBT∽△PTA (two angles are equal and two triangles are similar)

Then Pb: pt = pt: AP.

Namely: PT? =PB PA

secant theorem

Secant theorem: the product of the distance from two secant lines of a circle drawn from a point outside the circle to the intersection of each secant line and the circle is equal.

A straight line and an arc have two common points, so we say that this straight line is the secant of this curve.

Theorems related to secant include secant theorem and secant theorem. It is often used in questions about circles.

Similar to secant theorem: two secants intersect at point P, secant M intersects at point A 1 B 1, and secant N intersects at point A2 B2, then PA 1 pb 1 = PA2 pb2.

As shown in the figure, straight lines ABP and CDP are two secant lines of ⊙O drawn from point P, which proves that PA Pb = PC PD.

Proof: connect AD, BC≈A and

∠C is opposite to the arc BD.

On ∴ ∠DAP=∠BCP from the theorem of the angle of circle.

∫∠P = p again.

∴△ADP∽△CBP

(If two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar. )

∴AP:CP=DP:BP

That is, AP BP = CP DP.

Vertical chord theorem

The diameter perpendicular to the chord bisects the chord and the two arcs it faces.

Let it be ⊙O, DC is the diameter, AB is the chord, AB⊥DC is at point E, and AB intersects with CD at point E. It is proved that AE=BE, arc AC= arc BC, and arc AD= arc BD.

Connect OA and OB at point a and point b respectively.

∵OA and OB are the radii of∵ O.

∴OA=OB

△ OAB is an isosceles triangle.

∵AB⊥DC

∴AE=BE, ∠AOE=∠BOE (isosceles triangle with three lines in one)

∴ arc AD= arc BD, ∠AOC=∠BOC.

∴ arc AC= arc BC

Alternating line segment theorem

The tangential angle is equal to the corresponding circumferential angle. (Chord tangent angle is the angle between tangent and chord. )

It is known that the tangent circle O of the straight line PT is at point C, and BC and AC are the chords of the circle O. ..

Verification: ∠TCB= 1/2∠BOC=∠BAC.

Proof: Let the center of the circle be O and connect OC, OB,.

∠∠OCB =∠OBC

∴∠ocb= 1/2*( 180-∠ Bank of China)

∠∠BOC = 2∠BAC。

∴∠OCB=90 -∠BAC

∴∠BAC=90 -∠OCB

∠ TCB = 90-∠ OCB。

∴∠TCB= 1/2∠BOC=∠BAC

To sum up: ∠TCB= 1/2∠BOC=∠BAC

Equation of circle

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1, standard equation of circle:

In the plane rectangular coordinate system, the standard equation of a circle with point O(a, b) as the center and R as the radius is (x-a)2+(y-b)2=r2.

In particular, with the origin as the center, the radius is r (r >; 0) is x2+y2=r2.

2, the general equation of circle:

Equation x2+y2+Dx+Ey+F=0 can be transformed into (x+D/2)2+(y+E/2)2=(D2+E2-4F)/4. Therefore, there are:

(1), when D2+E2-4f >: 0, the equation represents a circle with (-D/2, -E/2) as the center and (D2+E2-4F)/2 as the radius;

(2) When D2+E2-4F=0, the equation represents a point (-D/2,-e/2);

③ When D2+E2-4f

3, the parameter equation of the circle:

The parametric equation of a circle with point O(a, b) as the center and R as the radius is x=a+r*cosθ, y=b+r*sinθ, where θ is the parameter.

Endpoint formula of circle:

If two points A (A 1, B 1) and B (A2, B2) are known, the line segment AB is taken as the diameter.

The equation of a circle is (x-a1) (x-a2)+(y-b1) (y-B2) = 0.

The eccentricity of a circle is e=0, and the radius of curvature of any point on the circle is r.

The tangent equation of a point M(a0, b0) passing through the circle x2+y2=r2 is A0 * x+B0 * y = R 2.

A point M(a0, b0) outside the circle (x2+y2=r2) leads to two tangents of the circle, which are A and B. Then the equation of the straight line where these two points are located is A0 * x+B0 * y = R 2.

4. The three-point equation of a circle: The equation of a circle passing through three non-collinear points A (X 1, Y 1), B (X2, Y2) and C (X3, Y3) is [2]?

Drawing mode

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Generally, you can draw a circle with a compass, or use a rope, one end of which is fixed on the ground and the other end is rotated, and the circle can be turned out. The longer the rope, the bigger the circle.

Draw a circle with AutoCAD

In AutoCAD "Drawing" drop-down menu, six drawing methods of "circle" are listed, which are briefly described as follows:

(1) Draw a circle with the center and radius: click the drawing command with the mouse, and then follow the prompts;

(2) Draw a circle with the center and diameter: click the drawing command with the mouse, and then follow the prompts;

(3) Use two points to determine the diameter and draw a circle: click the drawing command with the mouse, and then follow the prompts;

(4) Determine the diameter and draw a circle with three points: click the drawing command with the mouse and then follow the prompts;

(5) Draw a circle tangent to two graphic objects and having a certain radius: click the drawing command with the mouse, and then follow the prompts.

Richtext control draws a circle.

Defines an array to store one or more points.

Then it is realized according to the following steps

1 Generate a control (such as Label) and adjust the corresponding properties.

2. Create a temporary image in memory as a canvas, and use GDI+ and other drawings to draw the image on the canvas.

3. Set the attribute value of the generated control image or BackGroundImage to the image generated in step 2.

4 Use richtextbox1.controls.add method to add controls (you can specify their coordinates).

5 Record the coordinates of the currently added control in the array (for example, corresponding to the 1 th data).

6 add RichTextBox 1. Scroll event code, where,

Circular (2 sheets)

Calculate the position of adding the control by getting the value of the scroll bar.

Description: controls can be generated by code (recommended)

This method is different from RichTextBox method in QQ chat window circulating on the Internet.

Belong to simple type

When participating in scroll bar scrolling, you must define an array to reposition the target control.

Introduction to history

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A circle is a seemingly simple shape, but in fact it is very wonderful. The ancients first got the concept of circle from the sun and the moon on the fifteenth day of the lunar calendar. On the Neanderthal 18000 years ago, he used to drill holes in animal teeth, gravel and stone beads, some of which were very round. In the pottery age, many pottery were round. Round pottery is made by putting clay on a turntable. When people start spinning, they make round stone spindles or ceramic spindles. The ancients also found it easier to roll when carrying logs. Later, when they were carrying heavy objects, they put some logs under big trees and stones and rolled them around, which was of course much more labor-saving than carrying them.

About 6000 years ago, Mesopotamia made the world's first wheel-a round board. About 4000 years ago, people fixed round boards under wooden frames, which was the original car.

You can make a circle, but you don't necessarily know its nature. The ancient Egyptians believed that the circle was a sacred figure given by God. It was not until more than two thousand years ago that China's Mozi (about 468- 376 BC) gave the definition of a circle: a circle, a circle of equal length. It means that a circle has a center and the length from the center to the circumference is equal. This definition is 100 years earlier than that of the Greek mathematician Euclid (about 330 BC-275 BC).

The ratio of the circumference to the diameter of any circle is a fixed number. We call it pi, which is expressed by the letter π. It is an infinite acyclic decimal, π = 3. 14 15926535 ... but in practical application, it usually takes only an approximate value, namely π≈3. 14. If c is used to represent the circumference of a circle: C=πd or C=2πr, it is said in the weekly calculation book. When the Mesopotamians made the first wheel, they only knew that pi was 3. When Liu Hui annotated Nine Chapters Arithmetic in the Wei and Jin Dynasties in 263 AD, he found that "Three Circumferences and One Diameter" was only the ratio of the circumference to the diameter of a regular hexagon inscribed with a circle. He founded secant technology, and thought that when the number of inscribed sides of a circle increased infinitely, the circumference was closer to the circumference of a circle. He calculated the pi of a regular 3072 polygon inscribed in a circle = 3927/1250. Liu Hui applied the concept of limit to solving practical mathematical problems, which is also a great achievement in the history of mathematics in the world. Zu Chongzhi (AD 429-500) continued to calculate on the basis of predecessors' calculations, and found that pi was between 3. 14 15926 and 3. 14 15927, which was the earliest numerical value accurate to seven decimal places in the world. He also used two decimal values to express pi: 22/7 is called about. In Europe, it was not until 1000 years later16th century that the Germans Otto (A.D. 1573) and Antoine Z got this value. Nowadays, with electronic computers, pi has been calculated to five trillion digits after the decimal point.