1. Conditions for determining a circle: center → position, radius → size. 2. Concepts related to circle: chord diameter, arc semicircle, upper arc, lower arc, central angle, circumferential angle and chord center distance. 3. Symmetry of the circle: The circle is both an axisymmetric figure and a centrally symmetric figure. 4. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the two arcs opposite the chord. Inference: The diameter (not the diameter) of bisecting the chord is perpendicular to the chord and bisects the two arcs opposite the chord. 5. Relationship among central angle, arc, chord and chord center distance: In the same circle or within the same circle, the central angle of the circle has equal arc, chord and chord center distance. Extension: among these four groups, as long as one group is equal, the other groups are equal. 6. Theorem of circumferential angle: ① The circumferential angle is equal to half of the central angle of the same arc; (2) In the same circle or equal circle, the circumferential angle of the same arc or equal arc is equal, which is equal to half of the central angle of the arc; Equal circumferential angles are opposite to equal arcs; ③ The circumferential angle opposite to a semicircle (or diameter) is a right angle, and the chord opposite to a circumferential angle of 90 is a diameter. 7. Inner heart and outer heart: ① The inner heart is the intersection point of the bisector of the inner angle of the triangle, and its distance to the three sides of the triangle is equal. (2) The epicenter is the intersection of the perpendicular lines of the three sides of the triangle, and its distance to the three vertices of the triangle is equal. 8. positional relationship between straight line and circle: intersection → d r, tangency →d=r.9, judgment of tangency: "it is a little connected with the center of the circle" → proof of verticality. "Nothing is vertical" → Prove D = R. The property of tangent: the tangent of a circle is perpendicular to the radius passing through the tangent point. 10, tangent length theorem: two tangents of a circle are drawn from a point outside the circle, and their tangent lengths are equal. The connecting line between this point and the center of the circle bisects the included angle of the two tangents. 1 1, the properties of inscribed quadrangles: the diagonals of inscribed quadrangles are complementary, and each outer angle is equal to its inner diagonal. 12, the nature of the circumscribed quadrilateral: the sum of the opposite sides of the circumscribed quadrilateral is equal. 13, the positional relationship between circles: outward → D > R+R. External → D = R+R. Intersection → R-R < D < R+R. Internal → D < R-R. 14, positive. 15, arc length and sector area: L=nπR/ 180. S sector =nπR2/360. 16, lateral area and total area of cone: generatrix length of cone = radius of sector, circumference of cone bottom = arc length of sector, lateral area of cone = sector area, and total area of cone = sector area.