Summary of knowledge points in junior high school mathematics circle-1. Three points that are not on the same straight line determine a circle.
2. The vertical diameter theorem bisects the chord perpendicular to its diameter and bisects the two arcs opposite the chord.
Inference 1 ① bisect the diameter of the chord perpendicular to the chord (not the diameter) and bisect 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.
Inference 2 The arcs between two parallel chords of a circle are equal.
3. A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
4. A circle is a set of points whose distance from a fixed point is equal to a fixed length.
5. The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
6. The outside of a circle can be regarded as a collection of points whose center is farther than the radius.
7. The same circle or the same circle has the same radius.
8. The distance to a fixed point is equal to the trajectory of a fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.
9. Theorem In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.
10. It is inferred that in the same circle or equal circle, if one set of quantities in two central angles, two arcs, two chords or the chord-center distance between two chords is equal, the corresponding other set of quantities is also equal.
1 1 Theorem The diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal diagonal.
12.① intersection point d of straight line l and ⊙O
(2) the tangent of the straight line l, and ⊙ o d = r.
③ Lines L and ⊙O are separated from each other d>r.
13. The judgment theorem of tangent is that the outer end of the radius and the straight line perpendicular to this radius are the tangents of the circle.
14. The property theorem of tangent. The tangent of a circle is perpendicular to the radius passing through the tangent point.
15. Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
16. Inference 2 A straight line that crosses the tangent point and is perpendicular to the tangent must pass through the center of the circle.
17. The tangent length theorem leads to two tangents of a circle from a point outside the circle. Their tangents are equal in length, and the connecting line between the center of the circle and this point bisects the included angle between the two tangents.
18. The sum of two opposite sides of the circumscribed quadrangle of a circle is equal, and the outer angle is equal to the inner diagonal.
19. If two circles are tangent, then the tangent point must be on the line.
20.① Two circles are separated by d>. R+r ② Circumscribes two circles D = R+R.
③. Two circles intersect R-rr)
④ inscribed circle d = r-r (r >; R) (5) Two circles contain dr)
2 1. Theorem The intersection line of two circles bisects the common chord of two circles vertically.
22. Theorem divides a circle into n(n≥3):
(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.
(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.
23. Theorem Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
24. Each inner angle of a regular N-polygon is equal to (n-2) ×180/n.
25. Theorem The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
26. The area of a regular N-polygon Sn=pnrn/2 p represents the perimeter of the regular N-polygon.
27. The regular triangle area √3a/4 a indicates the side length.
28. If there are k positive N corners around a vertex, since the sum of these corners should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.
29. Calculation formula of arc length: L = nσR/ 180.
30. Sector area formula: s sector =n r 2/360 = LR/2.
3 1. Inner common tangent length = d-(R-r) Outer common tangent length = d-(R+r)
32. Theorem The angle of an arc is equal to half of its central angle.
33. Inference 1 is equal to the circumferential angle of the same arc or equal arc; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
34. Inference 2 The circumference angle (or diameter) of a semicircle is a right angle; A chord with a circumferential angle of 90 is a diameter.
35. the arc length formula l=a*r a is the radian number of the central angle r >; 0 sector area formula s= 1/2*l*r
Summary of knowledge points of junior middle school mathematics circle 2 1 and circle
Some properties of 1, circle
In a plane, the line segment OA rotates once around its fixed end point O, the figure formed by the rotation of the other end point A is called a circle, the fixed end point O is called a center, and the line segment OA is called a radius.
Judging from the meaning of the circle:
All points on a circle whose distance from a fixed point (center o) is equal to a fixed length are on the circle.
In other words, a circle is a set of points whose distance to a fixed point is equal to a fixed length, and the inside of a circle can be regarded as a circle. A set of points whose distance from the center is less than the radius.
The outside of a circle can be regarded as a collection of points whose distance from the center of the circle is greater than 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.
Divide the two endpoints of a circle with any diameter into two arcs, each arc is called a semicircle, and the arc larger than the semicircle is called an optimal arc; An arc smaller than a semicircle is called a bad arc. A circle composed of chords and opposing arcs is called an arch.
Two circles with the same center and different radii are called concentric circles.
Two circles that can overlap are called equal circles.
The same circle or the same circle has the same radius.
In the same circle or equal circle, arcs that can overlap each other are called equal arcs.
Second, the circle that passes three o'clock
L, a circle passing through three points
Practice of circle passing through three points: use the middle vertical line to find the center of the circle.
Theorem Three points that are not on the same straight line determine a circle.
The circle passing through each vertex of a triangle is called the circumscribed circle of the triangle, the center of the circumscribed circle is called the center of the circle, and this triangle is called the inscribed triangle of the circle.
Step 2 reduce to absurdity
Three steps of reduction to absurdity:
(1) The conclusion of the hypothetical proposition is not valid;
(2) Starting from this assumption, through reasoning and argumentation, the contradiction is obtained;
(3) It is concluded from the contradiction that the hypothesis is incorrect, thus affirming that the conclusion of the proposition is correct.
For example, verify that at most one angle in a triangle is obtuse.
Prove that there are more than two obtuse angles
Then sum of two obtuse angles >: 180
Contradicting that the sum of the internal angles of the triangle is equal to 180.
Obtuse angles cannot exceed two.
In other words, a person can play dumb at most.
Third, perpendicular to the diameter of the chord.
A circle is an axisymmetric figure, and every straight line passing through the center of the circle is its axis of symmetry.
Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord.
Reasoning 1: bisect the diameter of the chord perpendicular to the chord (not the diameter) and bisect the two arcs opposite to the chord.
The perpendicular bisector of the chord bisects the two arcs opposite the chord through the center of the circle.
Bisect the diameter of an arc opposite to the chord, bisect the chord vertically, and bisect another arc opposite to the chord.
Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.
Fourth, the relationship between central angle, arc, chord and chord center distance
A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
In fact, a circle can rotate at any angle around the center of the circle and coincide with the original figure.
The angle whose vertex is the center of the circle is called the central angle, and the distance from the center of the circle to the chord is called the chord distance.
Theorem: In the same circle or equal circle, the central angle of equal circle has equal arc, equal chord and equal chord distance.
Reasoning: In the same circle or the same circle, if one of two central angles, two arcs, two chords or the distance between two chords is equal, the other corresponding components are equal.
Verb (abbreviation for verb) Angle of circle.
The angle whose vertex is on the circle and whose two sides intersect the circle is called the circumferential angle.
Reasoning 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.
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.
Junior high school mathematics circle 3 knowledge points summary 1, symmetry:
A: The symmetry of a circle exists in some other figures, but the circle has countless symmetry axes, which are not found in other figures. The vertical diameter theorem, tangent length theorem and the calculation of positive N-polygon are all applied to this characteristic.
B: Rotation invariance, the relationship between central angle, arc, chord and chord center distance. When you encounter some exercises about circles, you should make full use of this feature and find solutions to many problems.
2. Three angles: the central angle, the outer angle and the outer angle (diagonal) of the quadrilateral inscribed by the circle. This is the basic method of finding angle equality in circle related problems.
3. Three sags: vertical diameter theorem, circumferential angle of diameter, and properties of tangent. It can effectively transform many problems into right triangles and solve them.
4. Four relationships: the positional relationship between a point and a circle, the positional relationship between a straight line and a circle, the positional relationship between a circle and a regular polygon, the determination and properties of a tangent and related calculations are the key points.
5. Related calculation problems: the calculation of line segments and regular polygons, the calculation of sector and shadow areas, and the calculation of side development diagrams of cylinders and cones.
6. The general method of adding auxiliary lines in a circle: adding auxiliary lines related to the vertical diameter theorem, adding auxiliary lines related to tangents (creating right-angle auxiliary lines), and adding auxiliary lines related to the quadrilateral inscribed in the circle; When two circles intersect, they are chords; when they are tangent, they are tangents. In short, when adding auxiliary lines, the basic graphics should be constructed and improved, and the integrity of the graphics should not be destroyed.