[On the basic properties and theorems of circles]

Determination of circle: three points that are not on the same straight line determine a circle.

Symmetry of circle: A 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 arc opposite to the chord. Inverse theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the arc opposite to the chord.

[On the Properties and Theorems of Central Angle and Central Angle]

In the same circle or in the same circle, if one group of two central angles, two peripheral angles, two arcs and two chords is equal, the corresponding other groups are equal respectively.

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

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

[On the properties and theorems of circumscribed circle and inscribed circle]

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; 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.

[On the Properties and Theorems of Tangents]

The tangent of the circle is perpendicular to the diameter of the tangent point; A straight line passing through one end of a diameter and perpendicular to the diameter is the tangent of the circle.

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.

The nature of the tangent: (1) The straight line perpendicular to this radius through 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: the length of two tangents from a point outside the circle to the circle, etc.

[About the Calculation Formula of Circle]

1. The circumference of the circle C=2πr=πd 2. The area of the circle S=πr? 0? 5 3. Sector arc length l=nπr/ 180

4. Sector area S=nπr? 0? 5/360=rl/2 5。 Transverse area of cone S=πrl.

Analytic geometric properties and theorems of circles

Analytic geometric equation of circle

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

General equation of a circle: expand the standard equation of a circle, shift the terms and merge the similar terms, and the general equation of a circle can be obtained as X 2+Y 2+DX+EY+F = 0. Compared with the standard equation, in fact, D=-2a, E=-2b and f = a 2+b 2.

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

[Judgment of the positional relationship between circle and straight line]

In the plane, the general method to judge the positional relationship between the straight line Ax+By+C=0 and the circle X 2+Y 2+DX+EY+F = 0 is:

1. From Ax+By+C=0, y = (-c-ax)/b, where b is not equal to 0, substitute x 2+y 2+dx+ey+f = 0, that is, it becomes a quadratic equation f(x)=0. Using the symbol of discriminant B 2-4ac, the positional relationship between a circle and a straight line can be determined as follows:

If b 2-4ac > 0, the circle and the straight line have two intersections, that is, the circle and the straight line intersect.

If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line.

If b 2-4ac

2. If B=0 indicates that the straight line is Ax+C=0, that is, X =-C/A, parallel to the Y axis (or perpendicular to the X axis), change X 2+Y 2+DX+EY+F = 0 to (X-A) 2+(Y-B) 2 = R, and let Y =

When x =-c/a X2, the straight line is out of the circle;

When x 1

Radius r, diameter d