2. The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.

3. The outside of a circle can be regarded as a collection of points whose center distance is greater than the radius.

4. The same circle or the same circle has the same radius.

5. The distance to the fixed point is equal to the trajectory of the fixed-length point, which is a circle with the fixed point as the center and the fixed length as the radius.

6. It is known that the locus of the points with the same distance between the two ends of a line segment is the middle vertical line of the line segment.

7. The locus of a point with equal distance to both sides of a known angle is the bisector of this angle.

8. The locus of a point with equal distance to two parallel lines is a straight line parallel to these two parallel lines and with equal distance.

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

10, the vertical diameter theorem bisects the chord perpendicular to the diameter of the chord and bisects the two arcs opposite the chord.

1 1, inference 1:

(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects 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.

12, Inference 2: The arcs sandwiched by two parallel chords of a circle are equal.

13. A circle is a centrosymmetric figure with the center of the circle as the symmetry center.

14, Theorem: In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.

15, inference: If one set of quantities in the same circle or equal circle, two central angles, two arcs, two chords or the distance between two chords are equal, the corresponding other set of quantities are also equal.

16, theorem: the angle of an arc is equal to half of its central angle.

17, inference: 1 has the same arc or equal arc with equal circumferential angle; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

18, inference: the circumferential angle of 2 semicircles (or diameters) is a right angle; A chord with a circumferential angle of 90 is a diameter.

19, 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.

Theorem: Diagonal lines of inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.

2 1, ① intersection of straight line l and ⊙O D < R

(2) the tangent of the straight line l, and ⊙ o d = r.

③ lines l and ⊙O are separated by d > r.

22. 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.

23. The nature of the tangent theorem The tangent of a circle is perpendicular to the radius passing through the tangent point.

24. Inference 1 A straight line passing through the center and perpendicular to the tangent must pass through the tangent point.

25. Inference 2 A straight line passing through the tangent point and perpendicular to the tangent line must pass through the center of the circle.

26. Tangent Length Theorem: Two tangents leading to a circle from a point outside the circle have the same tangent length, and the connecting line of this point bisects the included angle of the two tangents.

27. The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.

28. Chord tangent angle theorem: the chord tangent angle is equal to the circumferential angle of the arc pair it clamps.

29. Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.

30. Theorem of intersecting chords: The length of two intersecting chords in a circle divided by the product of the intersection point is equal.

3 1, inference: if the chord intersects the diameter vertically, then half of the chord is the average of the ratio of the two line segments formed by dividing it by the diameter.

32. Secant theorem: The tangent and secant of a circle are drawn from a point outside the circle, and the length of the tangent is the middle term in the ratio of the lengths of the two lines from this point to the intersection of the secant and the circle.

33. Inference: The product of two secant lines leading from a point outside the circle to the intersection of each secant line and the circle is equal.

34. If two circles are tangent, then the tangent point must be on the connecting line.

35.① the distance between two circles is d > r+r+r.

(2) circumscribed circle d d = r+r.

③ the intersection of two circles r-r < d < r+r (r > r).

④ inscribed circle d = r-r (r > r)

⑤ two circles contain d < r-r (r > r).

Theorem: The intersection line of two circles bisects the common chord of two circles vertically.

37. Theorem: Divide the 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.

Theorem: Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.

39. Each inner angle of a regular N-polygon is equal to (n-2) ×180/n.

40. Theorem: The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.

4 1, and the area Sn = PNRN/2 p of the regular N-polygon represents the perimeter of the regular N-polygon.

42, regular triangle area √ 3a/4a said side length.

43. If there are K positive N corners around a vertex, since the sum of these angles should be 360,

So k (n-2) 180/n = 360 is changed to (n-2)(k-2)=4.

44. Calculation formula of arc length: L = nσr/ 180.

45. Sector area formula: s sector =n r 2/360 = LR/2.

46. Inner common tangent length = d-(R-r) Outer common tangent length = d-(R+r)