1 Three points that are not on a straight line determine a circle.
The vertical diameter theorem divides the chord perpendicular to the chord diameter into two parts, and divides the two arcs opposite the chord into two parts.
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.
Inference 2 The arcs between two parallel chords of a circle are equal.
A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
A circle is a set of points whose distance from a fixed point is equal to a fixed length.
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.
7 The same circle or the same circle has the same radius.
The distance from the fixed point is equal to the trajectory of the fixed-length point, with the fixed point as the center and the fixed length as half.
Diameter circle
Theorem 9 In the same circle or in the same circle, the arcs with equal central angles are equal and the chords are equal.
Equal, the chord center distance of the opposite chord is equal.
10 Inference 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.
1 1 Theorem The inscribed quadrilateral of a circle is diagonally complementary, and any external angle is equal to it.
Internal diagonal of
12① 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 13 tangent is that 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 a 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 passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
17 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.
18 The sum of two opposite sides of the circumscribed quadrilateral of a circle is equal.
19 chord angle theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.
It is inferred that if the arc enclosed by two chord angles is equal, then the two chord angles are also equal.
30 Intersecting Chord Theorem The length of two intersecting chords and two straight lines in a circle divided by the product of the intersection points.
(to) equal to ...
3 1 It is inferred that if a chord intersects its diameter vertically, then half of the chord is composed of its diameter.
Proportional median of two line segments
The 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.
It is inferred that the product of two secants from a point outside the circle 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.
35① the circumscribed distance of two circles D > R+R ② the circumscribed distance 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 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.
If there are K positive N corners around a vertex, the sum of these corners should be
360, so k× (n-2) 180/n = 360 is changed to (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)
Theorem 47 The angle of a circle subtended by an arc is equal to half of its central angle.
48 Inference: 1 Same arc or equal arc with the same circumferential angle; 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; 90 degree circle angle
The chord on the right is the diameter.
50 sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.
5 1 cosine theorem b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..
52 The standard equation of a circle (x-a)2+(y-b)2=r2 Note: (A, B) is the coordinate of the center of the circle.
General equation x2+y2+Dx+Ey+F=0 for 53 circles Note: D2+E2-4f > 0
54 the arc length formula l=a*r a is the radian number of the central angle r >; 0 sector area formula s= 1/2*l*r 1 Three points that are not on the same line determine a circle.
The vertical diameter theorem divides the chord perpendicular to the chord diameter into two parts, and divides the two arcs opposite the chord into two parts.
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.
Inference 2 The arcs between two parallel chords of a circle are equal.
A circle is a central symmetrical figure with the center of the circle as the symmetrical center.
A circle is a set of points whose distance from a fixed point is equal to a fixed length.
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.
7 The same circle or the same circle has the same radius.
The distance from the fixed point is equal to the trajectory of the fixed-length point, with the fixed point as the center and the fixed length as half.
Diameter circle
Theorem 9 In the same circle or in the same circle, the arcs with equal central angles are equal and the chords are equal.
Equal, the chord center distance of the opposite chord is equal.
10 Inference 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.
1 1 Theorem The inscribed quadrilateral of a circle is diagonally complementary, and any external angle is equal to it.
Internal diagonal of
12① 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.
13 The judgment theorem that the tangent passes through the outer end of the radius and the straight line perpendicular to this radius is the tangent of a circle.
The property theorem of 14 tangent The tangent of a 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 passing through the tangent and perpendicular to the tangent must pass through the center of the circle.
17 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.
18 The sum of two opposite sides of the circumscribed quadrilateral of a circle is equal.
19 chord angle theorem The chord angle is equal to the circumferential angle of the arc pair it clamps.
It is inferred that if the arc enclosed by two chord angles is equal, then the two chord angles are also equal.
30 Intersecting Chord Theorem The length of two intersecting chords and two straight lines in a circle divided by the product of the intersection points.
(to) equal to ...
3 1 It is inferred that if a chord intersects its diameter vertically, then half of the chord is composed of its diameter.
Proportional median of two line segments
The 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.
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.
35① the circumscribed distance of two circles D > R+R ② the circumscribed distance 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 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.
If there are K positive N corners around a vertex, the sum of these corners should be
360, so k× (n-2) 180/n = 360 is changed to (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)
Theorem 47 The angle of a circle subtended by an arc is equal to half of its central angle.
48 Inference: 1 Same arc or equal arc with the same circumferential angle; 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; 90 degree circle angle
The chord on the right is the diameter.
50 sine theorem a/sinA=b/sinB=c/sinC=2R Note: where r represents the radius of the circumscribed circle of a triangle.
5 1 cosine theorem b2=a2+c2-2accosB Note: Angle B is the included angle between side A and side C..
52 The standard equation of a circle (x-a)2+(y-b)2=r2 Note: (A, B) is the coordinate of the center of the circle.
General equation x2+y2+Dx+Ey+F=0 for 53 circles Note: D2+E2-4f > 0
54 the arc length formula l=a*r a is the radian number of the central angle r >; 0 sector area formula s= 1/2*l*r
High school:
1. The properties and theorems of central angle and central angle;
In the same circle or in the same circle, if the distance between two central angles, two peripheral angles, two sets of arcs, two chords and one of the two chords is equal, the corresponding other groups are equal. An arc subtends a circumferential angle equal to half the central angle it subtends. The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter. If the length of an arc is twice that of another arc, then the angle of circumference and center it subtends is also twice that of the other arc.
Second, the properties and theorems of circumscribed circle and inscribed circle
① A triangle has a unique circumscribed circle and inscribed circle. The center of the circumscribed circle is the intersection of the perpendicular lines of each side of the triangle, and the distances to the three vertices of the triangle are equal;
(2) The center of the inscribed circle is the intersection of the bisectors of the inner angles of the triangle, and the distances to the three sides of the triangle are equal.
③R=2S△÷L(R: radius of inscribed circle, s: area of triangle, l: perimeter of triangle).
(4) the intersection of intersecting lines of two tangent circles (intersecting line: a straight line with two centers connected)
⑤ The midpoint m of PQ on the upper chord of circle O, and the intersection point m is defined as the intersection of two chords AB, CD, AD and BC with PQ on X and Y respectively, then M is the midpoint of XY.
3. If two circles intersect, the line segment connecting the centers of the two circles (or a straight line can be used) bisects the common chord vertically.
Fourth, the degree of the central angle is equal to the degree of the arc it faces.
The angle of a circle is equal to half the angle of the arc it faces.
6. The degree of the chord tangent angle is equal to half the degree of the arc it clamps.
Seven, the degree of the angle inside the circle is equal to half of the sum of the degrees of the arc opposite to this angle.
Eight, the degree of the outer angle of the circle is equal to half of the difference between the degrees of the two arcs cut at this angle.
Nine. Properties and Theorems of Tangents
The tangent of the circle is perpendicular to the radius of the tangent point; The straight line passing through one end of the radius and perpendicular to the radius is the tangent of the circle.
Judgment method of tangent: the straight line passing through the outer end of radius and perpendicular to this radius is the tangent of the circle.
The nature of the tangent: (1) The straight line perpendicular to this radius through the tangent point is the tangent of the circle. (2) The straight line perpendicular to the tangent point must pass through the center of the circle. (3) The tangent of the circle is perpendicular to the radius passing through the tangent point.
Tangent length theorem: the lengths of two tangents starting from a point outside the circle are equal, and the connecting line between this point and the center of the circle bisects the included angle of the tangents.
Attachment: [About the Calculation Formula of Circle]
1. circumference c = 2π r = π d.
2. The area of the circle s = π r 2;
3. Sector arc length l=nπr/ 180
4. Sector area S = (n π r 2)/360 = LR/2 (L is the arc length of the sector.
5. The lateral area of the cone is S=πrl 6. The central angle of the cone-side development diagram (sector) is n = 360 r/l (r is the radius of the bottom surface and l is the length of the bus).