The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
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.
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.
The locus of a point whose distance is equal to the two endpoints of a known line segment is the median vertical line of this line segment.
The locus from 7 to a point with equal distance on both sides of a known angle is the bisector of this angle.
The trajectory from 8 to the equidistant points of two parallel lines is a straight line parallel to and equidistant from these two parallel lines.
Theorem 9 Three points that are not on a straight line determine a circle.
10 vertical diameter theorem divides the chord perpendicular to its diameter into two parts, and divides the two arcs opposite to the chord into two parts.
1 1 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.
12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal.
13 circle is a symmetrical figure with the center of the circle as the symmetrical center.
Theorem 14 In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.
15 Inference: 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 equal.
Theorem 16 The angle of an arc is equal to half its central angle.
17 infers that 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.
18 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.
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 20 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.
The judgment theorem of tangent passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle.
The property theorem of tangent 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 of the circle and perpendicular to the tangent must pass through the tangent point.
Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
The tangent length theorem leads to two tangents of a circle from a point outside the circle, and their tangent lengths are equal. The line between the center of the circle and this point bisects the included angle of the two tangents.
The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.
28 Chord-tangent Angle Theorem Chord-tangent Angle is equal to the circumferential angle of the arc pair it clamps.
From this, it can be inferred that if the arc sandwiched by two chord angles is equal, then the two chord angles are also equal.
30 Intersecting Chord Theorem 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 proportional average of the two line segments formed by dividing it by the diameter.
The tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the middle term in the length ratio of the two lines at the intersection of this point and secant.
It is inferred that the product of two secant lines from a point outside the circle to the intersection of each secant line and the circle is equal.
If two circles are tangent, then the tangent point must be on the line.
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 36 The intersection of two circles bisects the common chord of two circles vertically.
Theorem 37 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.
Theorem 38 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
Each inner angle of a regular N-polygon is equal to (n-2) ×180/n.
Theorem 40 The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
The area of 4 1 regular n-polygon Sn = PNRN/2 p represents the perimeter of the regular n-polygon.
42 The area of a regular triangle √ 3a/4a indicates the side length.
43 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.
44 calculation formula of arc length: L=n r/ 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)