catalogue
Induction of circle knowledge points: the definition of circle.
Induction of circular knowledge points: each element of a circle.
Induction of circle knowledge points: the basic properties of circle.
Induction of circle knowledge points: the definition of circle.
1, a graph composed of points with a fixed point as the center and a fixed length as the radius.
2. A graph composed of points with the same distance to a fixed point on the same plane.
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Induction of circular knowledge points: each element of a circle.
1, radius: the line connecting a point on a circle with the center of the circle.
2. Diameter: Two points on the connecting circle have a line segment passing through the center of the circle.
3. Chord: a line segment connecting two points on a circle (the diameter is also a chord).
4. Arc: The curve between two points on a circle. A semicircle is also an arc.
(1) Bad arc: the arc is less than half a circle.
(2) Optimal arc: an arc larger than half a circle.
5. Central angle: an edge with the center of the circle as the vertex and the radius as the angle.
6. Circumferential angle: the vertex is on the circumference, and the two sides of the circumferential angle are chords.
7. Chord center distance: the length from the center of the chord to the vertical section of the chord.
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Induction of circle knowledge points: the basic properties of circle.
1, the symmetry of the circle.
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.
6. Let the radius of ⊙O be r and op = d. ..
7.( 1) The center of the circle crossing two points must be on the vertical line connecting the two points.
(2) Three points that are not on the same straight line determine a circle, the center of which is the intersection of the perpendicular lines of three sides, and the distances from this point to these three points are equal.
The outer center of a right triangle is the midpoint of the hypotenuse. )
8. The positional relationship between a straight line and a circle. D represents the distance from the center of the circle to a straight line, and r represents the radius of the circle.
A straight line and a circle have two intersections, and the straight line and the circle intersect; There is only one intersection point between a straight line and a circle, and the straight line is tangent to the circle;
There is no intersection between a straight line and a circle, but a straight line and a circle are separated.
9. In the plane rectangular coordinate system, A(x 1, y 1) and B(x2, y2). So AB=
10, tangent judgment of circle.
When (1)d=r, the straight line is the tangent of the circle.
The tangent point is not clear: draw a vertical line to prove the radius.
(2) The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.
Clear tangent point: uniform radius and vertical.
1 1, the properties of the tangent of the circle (supplementary).
(1) The diameter passing through the tangent point must be perpendicular to the tangent.
(2) A straight line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.
12, tangent length theorem.
(1) Tangent length: two tangents from a point outside the circle to the circle. The length of the connecting line between the tangent point and the point is called the tangent length of the point to the circle.
(2) Tangent length theorem.
PA and PB cut o at point a and point B.
PA=PB，2 .
13, inscribed circle and related calculation.
(1) The center of the inscribed circle of a triangle is the intersection of three internal bisectors, and its distances to the three sides are equal.
(2) As shown in the figure, in △ABC, AB=5, BC=6, AC=7, and ⊙O tangent △ABC is at points D, E and F.
Q: the length of AD, BE and cf.
Analysis: Let AD=x, then AD=AF=x, BD=BE=5-x, CE = CF = 7-X. 。
The equation can be obtained: 5-x+7-x=6, and the solution is x=3.
(3) At △ AB=c, C=90, AC=b, BC=a, AB=c.
Find the radius r of inscribed circle.
Analysis: Square ODCE was first proved,
Get CD=CE=r
AD=AF=b-r，BE=BF=a-r
b-r+a-r=c
Get r=
(4)S△ABC=
14, (supplementary)
(1) Chord angle: the vertex of the angle is on the circumference, one side of the angle is the tangent of the circle, and the other side is the chord of the circle.
As shown in the figure, BC intercepts ⊙O at point B, AB is the chord, ABC is the chord tangent angle, and ABC = D.
(2) Intersecting chord theorem.
The two chords AB and CD of the circle intersect at point P, then PAPB=PCPD.
(3) Cutting line theorem.
As shown in the figure, PA intercepts ⊙O at point A, and PBC is the secant of ⊙O, then PA2=PBPC.
(4) Inference: As shown in the figure, if PAB and PCD are secant of ⊙O, then PAPB=PCPD.
15, the positional relationship between circles.
(1) Extrapolation: dr 1+r2, with 0 intersection points;
Circumscribed: d=r 1+r2, with 1 intersections;
Intersection point: r 1-r2
Signature: d=r 1-r2, with 1 intersections;
Includes: 0d
(2) nature.
The intersection of two circles bisects the common chord vertically.
The straight line connecting two circles must pass through the tangent point.
16, calculation of correlation quantity in circle.
(1) The arc length is represented by L, the central angle is represented by N, and the radius of the circle is represented by R. ..
L=
(2) The area of the sector is represented by S. ..
S= S=+
(3) The lateral development of the cone is fan-shaped.
R is the radius of the bottom circle, and A is the length of the bus.
The central angle of the sector =
S side = ar S all = ar+ r2.
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