102 The interior of a circle can be regarded as a collection of points whose center distance is less than the radius.
The outer circle of 103 circle can be regarded as a collection of points whose center distance is greater than the radius.
104 The radius of the same circle or equal circle is the same.
The distance from 105 to the 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.
Theorem 109 Three points that are not on the same straight line determine a circle.
1 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 1 inference 1 ① bisect the diameter of the chord (not the diameter) perpendicular to the chord and bisect the two arcs opposite 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.
1 12 Inference 2 The arcs sandwiched by two parallel chords of a circle are equal.
1 13 circle is a centrosymmetric figure with the center of the circle as the symmetry center.
Theorem 1 14 In the same circle or in the same circle, arcs with equal central angles are equal, and chords with equal central angles are equal.
Equal, the chord center distance of the opposite chord is equal.
1 15 inference in the same circle or in the same circle, if two central angles, two arcs, two chords or two.
If one set of quantities in the chord-to-chord distance is equal, then the other sets of quantities corresponding to it are also equal.
Theorem 1 16 The angle of an arc is equal to half its central angle.
1 17 Inference 1 The circumferential angles of the same arc or the same arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.
1 18 Inference 2 The circumferential angle (or diameter) of a semicircle is a right angle; 90 degree circle angle
The chord on the right is the diameter.
1 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.
120 Theorem The inscribed quadrilateral of a circle is diagonally complementary, and any external angle is equal to it.
Internal diagonal of
12 1① the intersection of the straight line l and ⊙O is d < r.
(2) the tangent of the straight line l, and ⊙ o d = r.
③ lines l and ⊙O are separated by d > r.
122 tangent theorem The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.
123 The property theorem of tangent line The tangent line of a circle is perpendicular to the radius passing through the tangent point.
124 Inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.
125 Inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
126 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 between the two tangents.
127 The sum of two opposite sides of a circle's circumscribed quadrilateral is equal.
128 Chord Angle Theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.
129 Inference: If the arc enclosed by two chord tangent angles is equal, then the two chord tangent angles are also equal.
130 intersection chord theorem The product of two intersecting chords in a circle divided by the intersection point.
(to) equal to ...
13 1 Inference: If the chord intersects the diameter vertically, then half of the chord is formed by dividing it by the diameter.
Proportional median of two line segments
132 tangent theorem leads to the tangent and secant of a circle from a point outside the circle, and the tangent length is the point to be cut.
The proportional average of the lengths of two straight lines at the intersection of a straight line and a circle.
133 It is inferred that two secant lines of the circle are drawn from a point outside the circle, and the product of the lengths of the two lines from that point to the intersection of each secant line and the circle is equal.
134 If two circles are tangent, then the tangent point must be on the line.
135① perimeter of two circles D > R+R ② perimeter of two circles 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 136 The intersection of two circles bisects the common chord of two circles vertically.
Theorem 137 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 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 The radius and vertex of a regular N-polygon divide the regular N-polygon into 2n congruent right triangles.
The area sn = pnrn/2 p of the regular n-polygon represents the perimeter of the regular n-polygon.
The area of a regular triangle √ 3a/4a indicates the side length.
If there are K positive N corners around a vertex, since the sum of these angles should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.
Arc length calculation formula: L=n r/ 180.
Sector area formula: s sector =n r 2/360 = lr/2.
Inner common tangent length = d-(R-r) Outer common tangent length = d-(R+r)
The standard equation of a circle (x-a)2+(y-b)2=r2 Note: (A, B) is the center coordinate.
General equation of circle x2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0