Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite it. Inference 1 ① bisects the diameter of the chord (not the diameter) perpendicular to the chord, bisects the perpendicular bisector of two arcs opposite to the chord through the center of the circle, bisects the diameter of one arc opposite to the chord, bisects the chord vertically and bisects the other arc opposite to the chord, and infers that the arcs sandwiched by two parallel chords of circle 2 are equal.

Theorem? In the same circle or in the same circle, the arcs with the same central angles are equal, the chords are equal, and the chords are at the same distance.

10 inference? In the same circle or equal circle, if one of two central angles, two arcs, two chords or the chord-to-chord distance between two chords is equal, the corresponding other group is equal. Theorem 1 1? The diagonals of the inscribed quadrangles of a circle are complementary, and any external angle is equal to its internal angle.

12? ①? Does the straight line l intersect with ⊙O? dr

13 tangent theorem:? The straight line passing through the outer end of the radius and perpendicular to the radius is the tangent of the circle.

The property theorem of 14 tangent? The tangent of the 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 passes through the tangent and is perpendicular to the tangent must pass through the center of the circle.

17 tangent length theorem? Two tangents drawn from a point outside the circle are equal in length, and the connecting line between the center of the circle and the point bisects the included angle of the two tangents.

18 The sum of two opposite sides of the circumscribed quadrilateral of a circle is equal.

19 chord angle theorem? The chord tangent angle is equal to the circumferential angle of the arc pair it clamps.

20 inference? If the arc sandwiched between 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 the same.

3 1 inference? If the chord intersects the diameter vertically, then half of the chord is the proportional average of two line segments formed by its separate diameters.

32 cutting line theorem? The tangent and secant of a circle drawn from a point outside the circle, and the length of the tangent is the median term of the ratio of the lengths of the two lines from that point to the intersection of the secant and the circle.

33 Inference? Draw two secants of a circle 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 and the circle is equal.

If two circles are tangent, then the tangent point must be on the line. ①? Two circles apart? d>R+r？ ②? Are two circles circumscribed? d=R+r③？ Two circles intersect? R-rr)？ ⑤? The theorem that two circles contain d < r-r (r > r) 36? The line of intersection with two circles bisects the common chord of the two circles vertically. Theorem? Divide a circle into n(n≥3):

⑴? The polygon obtained by connecting these points in turn is the inscribed regular N-polygon of this circle. A polygon whose vertex is the intersection of adjacent tangents is a circumscribed regular polygon of a circle.

Theorem 38? 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.

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

4 1 area of regular n-polygon Sn = PNRN/2? P stands for the perimeter of a regular n-shape. 42. The area of a regular triangle √ 3a/4? A stands for side length.

43 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 is converted into (n-2)(k-2)=444. Arc length calculation formula: L=n/65438.

45 sector area formula: s sector = n r 2/360 = lr/246 internal common tangent length =? d-(R-r)？ Outer common tangent length =? d-(R+r)

Theorem 47? An arc subtends a circumferential angle equal to half the central angle it subtends.

48 inference 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.

49 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.

50 sine theorem? a/sinA=b/sinB=c/sinC=2R？ Attention:? Among them? r？ Represents the radius of the circumscribed circle of a triangle.

5 1 cosine theorem? b2=a2+c2-2accosB？ Note: Angle B is the standard equation of angle 52 circle between plane A and plane C? (x-a)2+(y-b)2=r2？ Note: (a, b) is the general equation of a circle with the center coordinate of 53? x2+y2+Dx+Ey+F=0？ Note: D2+E2-4f > 0

54 arc length formula? l=a*r？ A is the radian number r of the central angle? & gt0? Sector area formula? s= 1/2*l*r？