Outline of Mathematics Circle in Grade Three (Step by Step Learning)

The definition of 1. circle

There are two definitions of a circle:

First of all, a figure composed of all points whose distance from a plane to a fixed point is equal to a fixed length is called a circle.

Secondly, a line segment on the plane rotates 360 around its fixed endpoint O, and the trajectory left by its other end is called a circle.

2. Other related quantities of the circle

① Center and radius: (as defined) The fixed endpoint O is the center of the circle, which is represented by letters and marked as ⊙ o; The fixed length in the definition is the radius, which is represented by the letter r;

② Chord and diameter: the line segment connecting any two points on the circle is called chord, and the chord passing through the center of the circle is called diameter. The longest chord in a circle is the diameter;

③ Arc: The part between any two points on the circle is called arc, or arc for short. The arc larger than semicircle is called upper arc, and the arc smaller than semicircle is called lower arc;

4 central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle where the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle;

⑤ Equal circle: Two overlapping circles are called equal circles.

3. Vertical Diameter Theorem and Its Inference

① Theorem

If the diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and bisects the two arcs subtended by the chord.

② Inference (4)

Inference 1: the diameter (not the diameter) of the bisecting chord is perpendicular to this chord and bisects the two arcs opposite to this chord;

Inference 2: The perpendicular line of the chord passes through the center of the circle and bisects the two arcs opposite to the chord;

Inference 3: The diameter of the arc bisecting a chord bisects this chord vertically and bisects another arc opposite this chord.

Inference 4: In the same or equal circle, the arcs sandwiched by two parallel chords are equal.

4. Central angle and circumferential angle

(1) definition

① Central angle: the angle of the vertex at the center of the circle is called the central angle;

② Circumferential angle: the angle where the vertex is on the circle and the two sides intersect with the circle is called the circumferential angle.

(2) Theorem and inference

① Central angle

Theorem: In the same circle or in the same circle, equal central angles have equal arcs and equal chords.

Inference 1: If two arcs are equal in the same circle or equal circle, their central angles and chords are equal;

Inference 2: If two chords are equal in the same circle or in the same circle, then their central angles are equal and their arcs are equal.

② Circumferential angle

Theorem: In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc.

Inference 1: the circumference angle (or diameter) of a semicircle is a right angle, and the chord with a circumference angle of 90 is a diameter;

Inference 2: In the same circle or equal circle, if two circumferential angles are equal, the arcs they face must be equal;

Inference 3: Diagonal Complementarity of a quadrilateral inscribed in a circle.

5. The positional relationship between a point and a circle

The positional relationship between (1) point and circle

The positional relationship between a point and a circle is relatively simple and can be divided into three situations: inside the circle, on the circle and outside the circle.

Generally speaking, judging the positional relationship between a point and a circle is based on the distance between the point and the center of the circle and the radius of the circle. Suppose the radius of ⊙O is r, the distance between point P and the center of the circle is d, and the positional relationship between point P and ⊙O can be expressed as:

Point p is equivalent to d > r outside ⊙ o.

Point p is equivalent to d = r on ⊙ o.

Point p is equivalent to d < r in ⊙ o.

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

Three points that are not on the same straight line determine a circle. According to this theorem, we can make a circle through the three vertices of an arbitrary triangle, which is called the circumscribed circle of the triangle. The center of the circumscribed circle is the intersection of the perpendicular lines of the three sides of the triangle, which is called the center of the triangle.

(3) Reduction to absurdity

The conclusion is not drawn directly from the known proposition, but from the assumption that the conclusion of the proposition is not valid, which leads to contradictions, and the assumptions made by contradictions are incorrect, thus making the original proposition valid. This method of proof is called reduction to absurdity.

6. The positional relationship between a straight line and a circle

The positional relationship between a straight line and a circle can be divided into three types: intersection, tangency and separation, as follows:

(1) intersection

A straight line and a circle have two common points, so this straight line intersects with the circle, and this straight line is called the secant of the circle.

(2) Tangency

If there is only one common point between a straight line and a circle, the straight line is tangent to the circle, which is called the tangent of the circle, and the common point is called the tangent point.

(3) separation

That is, straight lines and circles have nothing in common.

Assuming that the radius ⊙O is r and the distance from the straight line L to the center of O is d, according to the above definition, we can get:

The intersection of straight lines l and ⊙O is equivalent to d < r.

The tangent of the straight line l to ⊙O is equivalent to d = r.

The separation of straight lines l and ⊙O is equivalent to d > r.

7. Tangent theorem

Definition of (1) tangent

If there is only one common point between a straight line and a circle, then the straight line is tangent to the circle, and the common point is the tangent point.

(2) Tangent judgment theorem

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

(3) Tangent property theorem

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

(4) Tangent length

The length of the line segment between a point outside the circle and the tangent point is called the tangent length of the point to the circle.

(5) Tangent length theorem

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

8. Triangular inscribed circle

The circle tangent to each side of a triangle is called the inscribed circle of the triangle, and the center of the inscribed circle is the intersection of bisectors of three angles of the triangle, which is called the center of the triangle. In addition, we need to know that the distance from the center of the triangle to the three sides of the triangle is equal, which is the radius of the inscribed circle of the triangle.

9. The positional relationship between circles

The positional relationship between circles is mainly divided into three types: separation, tangency and intersection, which are described as follows:

(1) separation

If two circles have nothing in common, then they are said to be separated; Separation can be divided into outer separation and inner inclusion. There is a special case of two circles, that is, two circles are concentric.

(2) Tangency

If two circles have only one common point, they are said to be tangent; Tangency can be divided into external tangent and internal tangent.

(3) Intersection

The intersection of two circles is relatively simple, that is, if two circles have two common points, they are said to intersect.

10. Regular polygons and circles

Let's review what a regular polygon is-a polygon with equal sides and angles. We call it a regular polygon.

Regular polygons are closely related to circles. 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.

The center of the circumscribed circle of a regular polygon is called the center of the regular polygon, the radius of the circumscribed circle is called the radius of the regular polygon, the central angle of each side of the regular polygon is called the central angle of the regular polygon, and the distance from the center to one side of the regular polygon is called the vertex of the regular polygon.

Or: in a plane, the line segment OA rotates once around its fixed end point O, and the figure formed by the other end point A is called a circle. The fixed endpoint o is called the center of the circle, and the line segment OA is called the radius.

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. Two endpoints of any diameter of a circle divide the circle into two arcs, and each arc is called a semicircle. Two circles that can overlap are called equal circles. In the same circle or equal circle, arcs that can overlap each other are called equal arcs.

A circle is an axisymmetric figure, and any straight line with a diameter is its axis of symmetry.

The diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.

The diameter (not the diameter) that bisects the chord is perpendicular to the chord and bisects the two arcs opposite the chord.

We call the angle of the vertex at the center of the circle the central angle.

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

If two arcs are equal in the same circle or equal circle, their central angles are equal and their chords are equal.

In the same circle or equal circle, if two chords are equal, then their central angles are equal and their arcs are equal.

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

In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half the central angle of the arc.

The circumference angle of a semicircle (or radius) is a right angle, and the chord with a circumference angle of 90 is a diameter.

If all vertices of a polygon are on the same circle, then the polygon is called a circle inscribed polygon and the circle is called a polygon circumscribed circle.

In the same circle or in the same circle, if two circumferential angles are equal, the arcs they face must be equal.

Diagonal complementarity of quadrilateral inscribed in a circle.

Point p is outside the circle -d > R point p is inside the circle -d = r point p is inside the circle -d

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

After the three vertices of a triangle, a circle can be made, which is called the circumscribed circle of the triangle. The center of the circumscribed circle is the intersection of the perpendicular lines of the three sides of the triangle, which is called the center of the triangle.

Lines and circles have two things in common. At this time, we say that this straight line intersects the circle, and this straight line is called the secant of the circle.

Straight lines and circles have only one thing in common. At this time, we say that this straight line is tangent to the circle, which is called the tangent of the circle, and this point is called the tangent point.

Lines and circles have nothing in common. At this time, we say that straight lines and circles are separate.

Lines l and o-d

Lines l and o are separated-d >; r

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

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

The length of the line segment between a point outside the circle and the tangent point is called the tangent length of the point to the circle.

Two circles are said to be separated if they have nothing in common. (Outer and Inner) If two circles have only one common point, they are said to be tangent. If these two circles have two points in common, they are said to intersect.

The distance between the centers of two circles is called the center distance.

We call the center of the circumscribed circle of a regular polygon as the center of the regular polygon, the radius of the circumscribed circle as the radius of the regular polygon, the central angle of each side of the regular polygon as the central angle of the regular polygon, and the distance from the center to one side of the regular polygon as the vertex of the regular polygon.

In a circle with radius r, since the arc length corresponding to the central angle of 360 is the circumference of the circle c = 2π r, the arc length corresponding to the central angle n is

nπR

L=—

180

The figure enclosed by the two radii forming the central angle and the arc opposite to the central angle is called a sector.

In a circle with radius r, the sector area corresponding to the central angle of 360 is the circle area s = π r? nπR？

S sector =—

360

We call the line segment connecting the vertex and any point on the circumference of the cone bottom as the generatrix of the cone.

36.

37.RT△ a+b-c

At r =-

38. 2 in any triangle