All the knowledge points in this chapter of ninth grade mathematics circle

Chapter IV: Circles

I. Review of knowledge

Circumference: C=2πr or C=πd, area of circle: S=πr? Calculation method of ring area: S=πR? -πr？ Or S=π(R? - r？ R is the radius of the big circle and R is the radius of the small circle.

Third, the main points of knowledge

First, the concept of circle

The concept of set form: 1, a circle can be regarded as a set of points whose distance to a fixed point is equal to a fixed length;

2. Outside the circle: it can be considered as a set of points whose distance to a fixed point is greater than a fixed length;

3. The interior of a circle can be regarded as a collection of points whose distance to a fixed point is less than a fixed length.

The concept of trajectory form:

1, circle: the locus of a point whose distance to a fixed point is equal to a fixed length is a circle with the fixed point as the center and the fixed length as the radius;

The fixed endpoint o is the center of the circle. A line segment connecting any two points on a circle is called a chord, and a chord passing through the center of the circle is called a diameter. The part between any two points on a circle is called an arc.

2. Median vertical line: the locus to the point with the same distance from both ends of the line segment is the median vertical line of this line segment;

3. The bisector of the angle: the locus to the point with equal distance on both sides of the angle is the bisector of the angle;

4. The locus of a point with equal distance to a straight line is: two straight lines are parallel to this straight line, and the distance to this straight line is equal to a fixed length;

5. The locus of a point with equal distance to two parallel lines is a straight line parallel to these two parallel lines and with equal distance to the two straight lines.

Second, the positional relationship between a point and a circle

1, the point is in the circle, and the point is in the circle;

2. Point on the circle, point on the circle;

3. The point is outside the circle, and the point is outside the circle;

Third, the positional relationship between a straight line and a circle

1, straight lines and circles do not intersect;

2. Tangents of straight lines and circles have intersections;

3. A straight line and a circle have two intersections;

Fourth, the positional relationship between circles.

External separation (Figure 1) has no intersection;

The circumscribed circle (Figure 2) has an intersection point;

The intersection (Figure 3) has two intersections;

The inner tangent (Figure 4) has an intersection point;

Containment (Figure 5)

There is no intersection

;

Five, vertical diameter theorem

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

Inference 1: (1) bisects the diameter of the chord perpendicular to the chord (not the diameter) and bisects the two arcs opposite to 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) bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.

The above * * * four theorems, referred to as the 2-push-3 theorem for short: of the five conclusions of this theorem, as long as we know two of them, we can deduce the other three conclusions, namely:

① is the diameter ②.

③

(4) arc (5) arc.

Any two conditions in this paper will lead to the other three conclusions.

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

Namely: in ⊙, ∵∧

Arc arc

Six, the central angle theorem

The angle from the vertex to the center of the circle is called the central angle.

Theorem of Central Angle: In the same circle or in the same circle, equal central angles have equal chords, equal arcs and equal chord spacing. This theorem is also called 1 push 3 theorem, that is, in the above four conclusions,

As long as we know that 1 is equal, we can deduce three other conclusions.

Namely: ①; ②;

③; ④ Arc arc

Seven, the circle angle theorem

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

1, theorem of circumferential angle: the circumferential angle of the same arc is equal to half the angle of the center of the circle it faces.

That is, ∫ sum is the central angle and circumferential angle of an arc.

∴

2. Inference of the theorem of circle angle;

Inference 1: the circumferential angles of the same arc or equal arc are equal; In the same circle or equal circle, the arc opposite to the equal circle angle is equal arc;

That is, in ⊙, ∵ and ∵ are right angles of a circle.

∴

Inference 2: the circumferential angle of a semicircle or diameter is a right angle; The angle of a circle is a right angle, the arc is a semicircle and the chord is a diameter.

That is, in ⊙, ∵ is the diameter or ∵.

It's the 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.

That is, at delta,

△ Is it a right triangle or

Note: this inference is actually the inference of rectangle in second-grade geometry: the inverse theorem that the median line on the hypotenuse in a right triangle is equal to half of the hypotenuse.

Eight, circle inscribed quadrilateral

Theorem of inscribed quadrilateral of a circle: The diagonals of inscribed quadrilateral of a circle are complementary, and the outer angle is equal to its inner diagonal.

That is to say, in ⊙,

A quadrilateral is an inscribed quadrilateral.

∴

Nine. Properties and Judgement Theorem of Tangent

(1) Judgment theorem of tangency: the straight line passing through the outer end of radius and perpendicular to radius is tangent;

Two conditions: through the outer radius and perpendicular to the radius, both of which are indispensable.

That is, through the outer end of the radius.

∴ is the tangent of ⊙.

(2) property theorem: the tangent line is perpendicular to the tangent point radius (as shown in the above figure)

Inference 1: A straight line passing through the center and perpendicular to the tangent must pass through the tangent point.

Inference 2: The straight line whose tangent point is perpendicular to the tangent line must pass through the center of the circle.

The above three theorems and inferences are also called two-deduction theorems:

Namely: ① passing through the center of the circle; ② Overtangent point; (3) Vertical tangent, knowing two of the three conditions can lead to the last one.

X tangent length theorem

Tangent length theorem;

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

That is: ∵, yes, two tangents.

∴

divide equally

XI。 Cyclic power theorem

(1) Chord intersection theorem: Two chords intersect in a circle, and the product of two line segments divided by the intersection point is equal.

That is, at ⊙, ∵ chord, intersection,

∴

(2) Inference: If the chord intersects the diameter vertically, then half of the chord is the proportional average of the two line segments formed by dividing it by the diameter.

That is to say, when the diameter is ⊙,

∴

(3) Tangent Theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines from this point to the intersection of the secant and the circle.

That is, in ⊙, ∵ is tangent and secant.

∴

(4) Secant Theorem: The product of two secant lines leading from a point outside the circle to the intersection of each secant line and the circle is equal (as shown in the above figure).

Namely: in ⊙, ∵, which is secant.

∴

Twelve. Cosine Theorem of Two Circles

Common chord theorem of circles: the straight line connecting the centers of two circles is vertical and bisects the common chord of the two circles.

As shown in the picture: perpendicular bisector.

That is, ∵⊙, ⊙ intersects at two points.

Vertical bisection

Thirteen, the common tangent of the circle

Formula for calculating the common tangent length of two circles;

(1) Common tangent length: medium,;

(2) External common tangent length: it is the difference of radius; Inner common tangent length: it is the sum of radii.

Fourteen Calculation of regular polygons in a circle