[Definition of Circle] Geometrically speaking, a figure composed of all points whose distance from a plane to a fixed point is equal to a fixed length is called a circle. A fixed point is called the center of the circle and a fixed length is called the radius. Trajectory theory: the trajectory of a moving point on a plane with a certain length as the center is called a circle. Set theory: The set of points whose distance to a fixed point is equal to a fixed length is called a circle. [Relative quantity of a circle] Pi: The ratio of the circumference to the diameter and length of a circle is called Pi, and its value is 3.14159265358979323846 …, which is usually expressed by π. In calculation, 3. 14 16 is often taken as its approximate value. Arc chord: the part between any two points on the circle is called arc, or simply arc. An arc larger than a semicircle is called an upper arc, and an arc smaller than a semicircle is called a lower arc. A line segment connecting any two points on a circle is called a chord. The chord passing through the center of the circle is called the diameter. Central angle and central angle: the angle of the vertex on the center of the circle is called the central angle. The angle at which the vertex is on the circumference and both sides intersect with the circle is called the circumferential angle. Inner and outer center: the circle passing through the three vertices of the triangle is called the circumscribed circle of the triangle, and its center is called the outer center of the triangle. A circle tangent to all three sides of a triangle is called the inscribed circle of the triangle, and its center is called the heart. Sector: On a circle, the figure enclosed by two radii and an arc is called a sector. The development diagram of the cone is a sector. The radius of this sector becomes the generatrix of the cone. 〕 Letter representation of correlation between circles 〕 Circle-⊙ Radius -R arc-⌒ Diameter -D fan arc length/conic generatrix -L circumference -C area-S 〕 Position relationship between circle and other figures 〕 Position relationship between circle and point: Take point P and circle O as an example (let p be a point, then PO is the distance from the point to the center), and p is in [ P on ⊙O，po = r； P is within ⊙O, and PO r；; AB is tangent to ⊙O, po = r；; AB intersects with ⊙O, and PO < R. There are five positional relationships between two circles: if there is nothing in common, one circle is called external separation outside the other circle, and it contains; If there is only one common point, a circle is called circumscribed by another circle and inscribed by another circle; There are two things in common called intersection. The distance between the centers of two circles is called the center distance. The radii of the two circles are R and R respectively, and R≥r, and the center distance is P: outward separation P > R+R; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; Contains p; 0, a circle and a straight line have two intersections, that is, a circle and a straight line intersect. If b 2-4ac = 0, the circle and the straight line have 1 intersections, that is, the circle is tangent to the straight line. If b 2-4ac 0 parabola standard equation Y2 = 2pxy2 =-2pxystraight prism lateral area S=c*h oblique prism lateral area S=c'*h right pyramid lateral area S= 1/2c*h' right pyramid lateral area S =1/2 (cl = pi (r H=2pi*h conic lateral area S = 1 formula V= 1/3*pi*r2h oblique prism volume V=S'L Note: where s' is the cross-sectional area, L is the volume formula of a long cylinder with sides V=s*h cylinder V=pi* R2h 10 1 A circle is a point set whose distance to a fixed point is equal to a fixed length 102 The inside of a circle can be regarded as a point set whose distance to the center is less than the radius 103 The outside of a circle can be regarded as a point set whose distance to the center is greater than the radius. 04 The radius of the same circle or an equal circle is equal to the locus of its points. The locus of a circle with a fixed radius 106 is the locus of a point with the same distance from the two endpoints of a known line segment, the locus of a point with the same distance from both sides of a known angle, the locus of a bisector of this angle 108 is the locus of a point with the same distance from two parallel lines, and the locus of a straight line parallel to these two parallel lines/. 1 10 Vertical Diameter Theorem bisects the chord perpendicular to the chord diameter and bisects the two arcs opposite to the chord11inference 1 ① bisects the diameter (not the diameter) of the chord perpendicular to the chord, and the midpoints of the two arcs opposite to the chord pass through the center of the circle, and the two arcs opposite to the chord. The perpendicular bisecting chord and bisecting another arc 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. 1 14 Theorem In the same circle or an equal circle, equal central angles have equal arcs and equal chords. The distance between chords of a pair of chords is equal. 1 15 It is inferred that in the same circle or the same circle, if the distances between two central angles, two arcs, two chords or two chords are equal, the corresponding other components are equal. 1 16 Theorem: The circumferential angle of an arc is equal to half of its central angle. In the same circle or equal circle, the arc opposite to the equal circle angle is also equal. 1 18 infers that 2 semicircles (or diameters) are right angles; The chord subtended by the circumferential angle of 90 is 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 the diagonal complement of the inscribed quadrilateral of the right triangle 120 theorem circle. And any outer angle is equal to the intersection point of the inner diagonal line 12 1① and ⊙O D < R2, and the tangent judgment theorem of ⊙O D = R3 and ⊙O D > R 122 passes through the outer end of the radius, and the straight line perpendicular to this radius is the tangent of the circle. 5438+024 Inference 1 A straight line passing through the center and perpendicular to the tangent must pass through the tangent point 125 Inference 2 A straight line passing through the tangent point and perpendicular to the tangent must pass through the center 126. Two tangents drawn from a point outside the circle have the same tangent length. 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 the circumscribed quadrangle of a circle is equal. The tangent angle theorem is equal to the circumferential angle of the arc pair it clamps. It is deduced that if the arcs sandwiched by two chord tangent angles are equal, then the two chord tangent angles are equal to the two intersecting chords in the chord theorem circle. The product of the length of two lines divided by the intersection is equal to 13 1. It is deduced that if the chord intersects the diameter vertically, then half of the chord is the tangent and secant of the circle, which is drawn by the middle term 132 according to the ratio of two line segments formed by a point outside the circle. The tangent length is the ratio of the lengths of two lines from this point to the intersection of the secant and the circle. 133 This item infers that two secant lines are drawn from a point outside the circle, and the product of the lengths of the two lines from this point to the intersection of each secant line and the circle is equal to 134. If two circles are tangent, then the tangent point must be on the line 135① two circles are tangent to D > R+R ② two circles are tangent to d=R+r ③ two circles intersect R-R < D+R (R > R) ④ two circles are inscribed with D = R-R (R > R) ⑤ two circles contain D < R. Chord theorem 137 divides a circle into n (n ≥ 3): (1) The polygon obtained by connecting all points in turn is a regular n polygon inscribed in the circle; (2) Tangents that circle all points, and polygons whose vertices are the intersections of adjacent tangents are circumscribed regular N polygons of the circle; (138) Theorem Any regular polygon has a circumscribed circle and an inscribed circle, and these two circles are concentric respondents: handsome vultures.