The (1) circle is an axisymmetric figure, and its symmetry axis is the straight line where the diameter lies.

(2) A circle is a figure with a symmetrical center, and its symmetrical center is the center of the circle.

(3) A circle is a rotationally symmetric figure.

2. Vertical diameter theorem.

(1) bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.

(2) Inference:

Bisect the diameter (non-diameter) of a chord, perpendicular to the chord and bisecting the two arcs opposite the chord.

Bisect the diameter of the arc and bisect the chord of the arc vertically.

3. The degree of the central angle is equal to the degree of the arc it faces. The degree of the circle angle is equal to half the radian it subtends.

(1) The circumferential angles of the same arc are equal.

(2) The circumferential angle of the diameter is a right angle; The angle of a circle is a right angle, and the chord it subtends is a diameter.

4. In the same circle or equal circle, as long as one of the five pairs of quantities, namely two chords, two arcs, two circumferential angles, two central angles and the distance between the centers of two chords, is equal, the other four pairs are also equal.

5. The two arcs sandwiched between parallel lines are equal.

develop

First, the related concepts of circle

1, the definition of a circle

In each plane, the line segment OA rotates around its fixed end point O, the figure formed by the rotation of the other end point A is called a circle, the fixed end point O is called a center, and the line segment OA is called a radius.

2. The relationship between a straight line and a circle.

1. When there is only one * * * between the straight line and the circle, click to call it straight and cut the circle line.

2. The circle connected by triangle is called the outer center of the three-center angle.

3. The chord tangent angle is equal to the central angle of the clamping arc.

4. The circle tangent to the internal shape of the triangle is called the center of the three-center angle.

5. The semi-straight line perpendicular to the diameter must be the tangent of the circle.

6. A straight line that passes through a point outside half the diameter and ends vertically at that point is a tangent to a circle.

7. The semi-straight line perpendicular to the diameter is the tangent of the circle.

8. The tangent of the circle is perpendicular to the radius of the point.

3, the geometric representation of the circle

The circle centered on point O is marked as "⊙O" and pronounced as "circle O"

Second, the vertical diameter theorem and its inference

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

Inference 1:

(1) bisects the diameter of the chord (not the diameter) perpendicular to the chord and bisects the two arcs opposite the chord.

(2) The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord.

(3) The diameter of the arc bisecting the chord bisects the chord vertically and bisects another arc opposite the chord.

Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.

The vertical diameter theorem and its inference can be summarized as follows:

Over the center of the circle

Perpendicular to the chord

The diameter bisects the chord to know two and push three.

The best arc to bisect the chord.

A lower arc that is split in two by a chord.

Three, chord, arc and other definitions related to the circle

1, string

A line segment connecting any two points on a circle is called a chord. (AB in the figure)

2. Diameter

The chord passing through the center of the circle is called the diameter. (such as the upcoming CD)

The diameter is equal to twice the radius.

3, semicircle

The two endpoints of a circle with any diameter are divided into two arcs, and each arc is called a semicircle.

4, arc, arc, arc.

The part between any two points on a circle is called an arc.

The arc is represented by the symbol "⌒", and the arc with A and B as endpoints is marked as "",which is pronounced as "arc AB" or "arc AB".

An arc larger than a semicircle is called an optimal arc (usually represented by three letters); An arc smaller than a semicircle is called a bad arc (usually represented by two letters).

Fourth, the symmetry of the circle.

1, the axis symmetry of the circle

A circle is an axisymmetric figure, and every straight line passing through the center of the circle is its axis of symmetry.

2. The center of the circle is symmetrical

A circle is a central symmetrical figure with the center of the circle as the symmetrical center.

5. Theorem of the relationship between arc, chord, chord center distance and central angle.

1, central angle

The angle of the vertex at the center of the circle is called the central angle.

2, chord center distance

The distance from the center of the circle to the chord is called the chord center distance.

3. Theorem of the relationship between arc, chord, chord center distance and central angle.

In the same circle or in the same circle, the arcs with equal central angles are equal, the chords are equal, and the chord distance is equal.

Inference: In the same circle or equal circle, if one set of quantities in two circles, two arcs, two chords' central angles or two chords' central distances are equal, the corresponding other set of quantities are equal respectively.

Six, the theorem of circle angle and its inference

1, circle angle

The angle whose vertex is on the circle and whose two sides intersect the circle is called the circumferential angle.

2. The theorem of circle angle

An arc subtends a circumferential angle equal to half the central angle it subtends.

Inference 1: the circumferential angles of the same arc or equal arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

Inference 2: the circumferential angle of a semicircle (or diameter) is a right angle; A chord with a circumferential angle of 90 is a diameter.

Inference 3: If the median line of one side of a triangle is equal to half of this side, then this triangle is a right triangle.

Seven, the position of the point and the circle.

Let the radius of ⊙O be r and the distance from point P to the center of O be d, then there are:

d

D = ⊙ o on point p;

D>r point P is outside ⊙ O.

Eight, three o'clock, one lap.

1, a circle passing through three points

Three points that are not on the same straight line determine a circle.

2. The circumscribed circle of a triangle

A circle passing through the three vertices of a triangle is called the circumscribed circle of the triangle.

3. The outer center of the triangle

The center of the circumscribed circle of a triangle is the intersection of the perpendicular lines of the three sides of the triangle, which is called the center of the triangle.

4. Quadrilateral properties of inscribed circle (judging conditions of four-point * * * circle)

Diagonal complementarity of quadrilateral inscribed in a circle.

Nine, reduce to absurdity

First, assume that the conclusion in the proposition is not valid, and then through reasoning, lead to contradictions and judge that the hypothesis is incorrect, so as to get the original proposition to be valid. This method of proof is called reduction to absurdity.

X. positional relationship between straight line and circle

There are three positional relationships between a straight line and a circle, as shown below:

(1) intersection: when a straight line and a circle have two common points, it is called the intersection of the straight line and the circle. At this time, the straight line is called the secant of the circle, and the common point is called the intersection point;

(2) Tangency: When a straight line and a circle have only one common point, the straight line is said to be tangent to the circle, and the straight line is said to be tangent to the circle.

(3) Separation: When the straight line and the circle have nothing in common, it is called separation of the straight line and the circle.

If the radius ⊙O is r and the distance from the center o to the straight line l is d, then:

The straight line l intersects with O d

The straight line l is tangent to ⊙O, and d = r；;

The straight line l is separated from ⊙O d >; r；

XI。 Determination and properties of tangent line

1, tangent theorem

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

2. The property theorem of tangent line

The tangent of a circle is perpendicular to the radius passing through the tangent point.

Twelve. Tangent length theorem

1, tangent length

On the tangent of a circle passing through a point outside the circle, the length of the line segment between the point and the tangent point is called the tangent length from the point to the circle.

2. Tangent length theorem

Two tangents drawn from a point outside the circle are equal in length, and the connecting line between the center of the circle and the point bisects the included angle of the two tangents.

Thirteen, the position relationship between the circle and the circle

1, the positional relationship between circles

If there is nothing in common between two circles, then they are said to be separated, and separation can be divided into external and internal.

If two circles have only one common point, they are said to be tangent, and tangency can be divided into circumscribed and inscribed.

If two circles have two common points, they are said to intersect.

2. Center distance

The distance between two centers is called the distance between two centers.

3. The nature and judgment of the relationship between circles.

Let the radii of two circles be r and r, respectively, and the distance between the centers be d, then

D & gtR+r

Circumscribed circle d=R+r

The intersection of two circles R-r

The inscribed circle d = r-r (r >); r)

Two circles contain dr)

4. Important properties of tangency and intersection of two circles

If two circles are tangent, then the tangent point must be on the connecting line, they are axisymmetric figures, and the symmetry axis is the connecting line of two circles; The intersection of two circles bisects the common chord of the two circles vertically.

Fourteen, inscribed circle of triangle

1, inscribed circle of triangle

A circle tangent to all sides of a triangle is called the inscribed circle of the triangle.

2. The center of the triangle

The center of the inscribed circle of a triangle is the intersection of the bisectors of three internal angles of the triangle, which is called the' heart' of the triangle.

15. Concepts related to regular polygons

1, the center of the regular polygon

The center of the circumscribed circle of a regular polygon is called the center of this regular polygon.

2. Radius of regular polygon

The radius of the circumscribed circle of a regular polygon is called the radius of this regular polygon.

3. Vertices of regular polygons.

The distance from the center of a regular polygon to one side of the regular polygon is called the vertex of the regular polygon.

4. Central angle

The central angle of the circumscribed circle opposite to each side of a regular polygon is called the central angle of this regular polygon.

Sixteen, regular polygons and circles

1, the definition of regular polygon

Polygons with equal sides and angles are called regular polygons.

2. The relationship between regular polygon and circle

As long as a circle is divided into equal arcs, the inscribed regular polygon of this circle can be made, and this circle is the circumscribed circle of this regular polygon.

17. Symmetry of regular polygons

1, the regular polygon is symmetrical.

Regular polygons are all axisymmetric figures. A regular n-polygon has n symmetry axes, and each symmetry axis passes through the center of the regular n-polygon.

2. Central symmetry of regular polygon

A regular polygon with an even number of sides is a central symmetric figure, and its symmetric center is the center of the regular polygon.

3. Drawing regular polygons

First divide the circle into equal parts with a protractor or ruler, and then make a regular polygon.

Eighteen, arc length and sector area

1, arc length formula

The formula for calculating the arc length l corresponding to the central angle n is

2. Sector area formula

Where n is the degree of central angle of the sector, r is the radius of the sector, and l is the arc length of the sector.

3. The transverse area of the cone

Where l is the generatrix length of the cone and r is the grounding radius of the cone.

Problem-solving skills of mathematical circle problem in junior middle school

Calculation of radius and chord length, the distance from the chord center to the intermediate station.

If there are all lines on the circle, the radius of the center of the tangent point is connected.

Pythagorean theorem is the most convenient for the calculation of tangent length.

To prove that it is tangent, carefully distinguish the radius perpendicular.

Is the diameter, in a semicircle, to connect the chords at right angles.

An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely.

There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected.

Find tangent chord, same arc diagonal, etc.

If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides.

Also make an inscribed circle, and the bisector of the inner corner is a dream circle.

If you meet an intersecting circle, don't forget to make it into a string.

Two circles tangent inside and outside pass through the common tangent of the tangent point.

If you add a connector, the tangent point must be on the connector.

Adding a circle to the equilateral angle makes it not so difficult to prove the problem.

The auxiliary line is a dotted line, so be careful not to change it when drawing.

If the graph is dispersed, rotate symmetrically to carry out the experiment.