1 Definition of a circle: (There are two definitions of a circle)

2 inside and outside the circle

The positional relationship between three points and a circle:

① point d outside the circle > R2 point d on the circle = R3 point d inside the circle < R.

Concepts related to circle: chord, diameter, arc, semicircle, upper arc, lower arc, bow, concentric circle, equal circle and equal arc.

14. Pass through a three-point circle

Theorem 1: Three points that are not on the same line determine a circle.

2 The concepts of circumscribed circle of triangle, outer center of triangle and inscribed triangle of circle.

The definition of reduction to absurdity and the general steps of proving propositions by reduction to absurdity.

15. Diameter perpendicular to the chord

Axisymmetry of 1 circle

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

3 circle rotation invariance

16, the concept of central angle and chord center distance.

The relationship among central angle, arc, chord and chord center distance.

The relationship between the degree of the central angle and the degree of the arc it faces: the degree of the central angle is equal to the degree of the arc it faces

17, circle angle

The concept of 1 fillet

Theorem of circle angle: the circle angle of an arc is equal to half of the central angle of an arc.

3 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 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.

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

1, the definition of the positional relationship between a straight line and a circle and related concepts.

(1) When a line and a circle have two common points, it is called the intersection of the line and the circle, and then the line is called the secant of the circle.

Common points are called intersections.

(2) 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.

The common point is called the tangent point.

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

2. The nature and judgment of the position relationship between straight line and circle.

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

The intersection point d of straight lines l and o < r

Tangency of straight line l and o d=r

Lines l and ⊙O are separated from each other, and d > r.

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

3, the method of determining the tangent of the circle

Definition: A straight line with only one common point with a circle is the tangent of the circle;

Quantitative relationship: the straight line whose distance from the circle is equal to the radius is the tangent of the circle;

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

4. Property theorem: the tangent of a circle is perpendicular to the radius passing through the tangent point.

5. The inscribed circle of a triangle

Concepts such as inscribed circle of triangle:

The circle tangent to each side of a triangle is called the inscribed circle of the triangle, the center of the inscribed circle is called the heart of the triangle, and this triangle is called the circumscribed triangle of the circle.

A circle tangent to all sides of a polygon is called the inscribed circle of the polygon, and this polygon is called the circumscribed polygon of the circle.