Similar triangles's Judgment Theorem 1 Two angles are equal and two triangles are similar (ASA)
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)
Theorem If the hypotenuse of a right-angled triangle and one right-angled side and another right-angled side
The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.
The property theorem 1 similar triangles corresponds to the height ratio, and the ratio corresponding to the center line is equal to the corresponding angle.
The ratio of dividing lines is equal to the similarity ratio.
Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.
Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.
Chapter III Circle
A circle is a set of points whose distance from a fixed point is equal to a fixed length.
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.
105 The distance from the fixed point is equal to the trajectory of the fixed point, with the fixed point as the center, and the fixed length is half.
Diameter circle
106 and it is known that the locus of the point with the same distance between the two endpoints of the line segment is perpendicular to the line segment.
bisector
The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.
The trajectory from 108 to the point with the same distance from two parallel lines is parallel to these two parallel lines with a distance of.
A straight line of equality
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 chords are triangles with similar diameters. Decision Theorem 1 Two angles are equal and two triangles are similar (ASA)
Two right triangles divided by the height on the hypotenuse are similar to the original triangle.
Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).
Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS)
Theorem If the hypotenuse of a right-angled triangle and one right-angled side and another right-angled side
The hypotenuse of an angle is proportional to a right-angled side, so two right-angled triangles are similar.
The property theorem 1 similar triangles corresponds to the height ratio, and the ratio corresponding to the center line is equal to the corresponding angle.
The ratio of dividing lines is equal to the similarity ratio.
Property Theorem 2 The ratio of similar triangles perimeter is equal to similarity ratio.
Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.
Chapter III Circle
A circle is a set of points whose distance from a fixed point is equal to a fixed length.
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.
105 The distance from the fixed point is equal to the trajectory of the fixed point, with the fixed point as the center, and the fixed length is half.
Diameter circle
106 and it is known that the locus of the point with the same distance between the two endpoints of the line segment is perpendicular to the line segment.
bisector
The locus from 107 to a point with equal distance on both sides of a known angle is the bisector of this angle.
The trajectory from 108 to the point with the same distance from two parallel lines is parallel to these two parallel lines with a distance of.
A straight line of equality
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 equal circle, the arc opposite to the equal circle angle also
A preliminary understanding of the accusation circle
I. Definition of Circle and Related Quantity of Circle (28)
1. A graph 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.
2. 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.
3. 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.
The circle passing through the three vertices of a 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.
5. There are three positional relationships between a straight line and a circle: there is no separated common point; There are two common * * * points intersecting; A circle and a straight line have a unique common tangent point. This straight line is called the tangent of the circle, and this unique common point is called the tangent point.
6. There are five kinds of positional relations between two circles: if there is nothing in common, one circle is called external separation from the other, and it is called internal separation; 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.
7. 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.
Second, the letter representation of the circle (7)
Circle-⊙ radius -R arc-⌒ diameter-D.
Sector arc length/conical generatrix -l circumference -c area -s
Three. Basic Properties and Theorems of Circle (27)
1. Position relationship between POint p and circle o (let p be a point, then po is the distance from the point to the center of the circle):
P outside ⊙O, po > r；; P on ⊙O，po = r； P is within ⊙O, and po < r.
2. A circle is an axisymmetric figure, and its axis of symmetry is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.
3. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord. Inverse theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the arc opposite to the chord.
4. In the same circle or in the same circle, if one group of two central angles, two peripheral angles, two arcs and two chords is equal, the corresponding other groups are equal respectively.
5. The angle of a circle subtended by an arc is equal to half of the central angle subtended by it.
6. The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.
7. Three points that are not on the same straight line determine a circle.
8. 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; 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.
9. The positional relationship between straight line AB and circle O (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle):
AB and ⊙O are separated, po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
10. The tangent of the circle is perpendicular to the diameter of the tangent point; A straight line passing through one end of a diameter and perpendicular to the diameter is the tangent of the circle.
1 1. The positional relationship between circles (let the radii of two circles be r and r respectively, and R≥r and the center distance be p);
Exogenous p > r+r; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains P r；; P on ⊙O，po = r； P is within ⊙O, and po < r.
2. A circle is an axisymmetric figure, and its axis of symmetry is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.
3. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord. Inverse theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the arc opposite to the chord.
4. In the same circle or in the same circle, if one group of two central angles, two peripheral angles, two arcs and two chords is equal, the corresponding other groups are equal respectively.
5. The angle of a circle subtended by an arc is equal to half of the central angle subtended by it.
6. The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.
7. Three points that are not on the same straight line determine a circle.
8. 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; 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.
9. The positional relationship between straight line AB and circle O (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle):
AB and ⊙O are separated, po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
10. The tangent of the circle is perpendicular to the diameter of the tangent point; A straight line passing through one end of a diameter and perpendicular to the diameter is the tangent of the circle.
1 1. The positional relationship between circles (let the radii of two circles be r and r respectively, and R≥r and the center distance be p);
Exogenous p > r+r; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains P r；; P on ⊙O，po = r； P is within ⊙O, and po < r.
2. A circle is an axisymmetric figure, and its axis of symmetry is any straight line passing through the center of the circle. A circle is also a central symmetric figure, and its symmetric center is the center of the circle.
3. Vertical diameter theorem: the diameter perpendicular to the chord bisects the chord and bisects the arc opposite to the chord. Inverse theorem: bisecting the diameter of a chord (not the diameter) is perpendicular to the chord and bisecting the arc opposite to the chord.
4. In the same circle or in the same circle, if one group of two central angles, two peripheral angles, two arcs and two chords is equal, the corresponding other groups are equal respectively.
5. The angle of a circle subtended by an arc is equal to half of the central angle subtended by it.
6. The circumferential angle of the diameter is a right angle. The chord subtended by a 90-degree circle angle is the diameter.
7. Three points that are not on the same straight line determine a circle.
8. 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; 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.
9. The positional relationship between straight line AB and circle O (let OP⊥AB be in P, then PO is the distance from AB to the center of the circle):
AB and ⊙O are separated, po > r；; AB is tangent to ⊙O, po = r；; AB and ⊙O intersect, po < r.
10. The tangent of the circle is perpendicular to the diameter of the tangent point; A straight line passing through one end of a diameter and perpendicular to the diameter is the tangent of the circle.
1 1. The positional relationship between circles (let the radii of two circles be r and r respectively, and R≥r and the center distance be p);
Exogenous p > r+r; Circumscribed p = r+r; Intersection r-r < p < r+r; Inner cut p = r-r; It contains p.