First, the basic knowledge

I number and algebra a, number and formula: 1, rational number rational number: ① integer → positive integer /0/ negative integer ② score → positive fraction/negative fraction.

Number axis: ① Draw a horizontal straight line, take a point on the straight line to represent 0 (origin), select a certain length as the unit length, and specify the right direction on the straight line as the positive direction to get the number axis. ② Any rational number can be represented by a point on the number axis. (3) If two numbers differ only in sign, then we call one of them the inverse of the other number, and we also call these two numbers the inverse of each other. On the number axis, two points representing the opposite number are located on both sides of the origin, and the distance from the origin is equal. The number represented by two points on the number axis is always larger on the right than on the left. Positive numbers are greater than 0, negative numbers are less than 0, and positive numbers are greater than negative numbers.

Absolute value: ① On the number axis, the distance between the point corresponding to a number and the origin is called the absolute value of the number. (2) The absolute value of a positive number is itself, the absolute value of a negative number is its reciprocal, and the absolute value of 0 is 0. Comparing the sizes of two negative numbers, the absolute value is larger but smaller.

Operation of rational numbers: addition: ① Add the same sign, take the same sign, and add the absolute values. ② When the absolute values are equal, the sum of different symbols is 0; When the absolute values are not equal, take the sign of the number with the larger absolute value and subtract the smaller absolute value from the larger absolute value. (3) A number and 0 add up unchanged.

Subtraction: Subtracting a number equals adding the reciprocal of this number.

Multiplication: ① Multiplication of two numbers, positive sign of the same sign, negative sign of different sign, absolute value. ② Multiply any number by 0 to get 0. ③ Two rational numbers whose product is 1 are reciprocal.

Division: ① Dividing by a number equals multiplying the reciprocal of a number. ②0 is not divisible.

Power: the operation of finding the product of n identical factors A is called power, the result of power is called power, A is called base, and N is called degree.

Mixing order: multiply first, then multiply and divide, and finally add and subtract. If there are brackets, calculate first.

2. Real irrational numbers: Infinitely circulating decimals are called irrational numbers.

Square root: ① If the square of a positive number X is equal to A, then this positive number X is called the arithmetic square root of A. If the square of a number X is equal to A, then this number X is called the square root of A. (3) A positive number has two square roots /0 square root is 0/ negative number without square root. (4) Find the square root of a number, which is called the square root, where a is called the square root.

Cubic root: ① If the cube of a number X is equal to A, then this number X is called the cube root of A. ② The cube root of a positive number is positive, the cube root of 0 is 0, and the cube root of a negative number is negative. The operation of finding the cube root of a number is called square root, where a is called square root.

Real numbers: ① Real numbers are divided into rational numbers and irrational numbers. ② In the real number range, the meanings of reciprocal, reciprocal and absolute value are exactly the same as those of reciprocal, reciprocal and absolute value in the rational number range. ③ Every real number can be represented by a point on the number axis.

3. Algebraic expressions

Algebraic expression: A single number or letter is also an algebraic expression.

Merge similar items: ① Items with the same letters and the same letter index are called similar items. (2) Merging similar items into one item is called merging similar items. (3) When merging similar items, we add up the coefficients of similar items, and the indexes of letters and letters remain unchanged.

4. Algebraic expressions and fractions.

Algebraic expression: ① The algebraic expression of the product of numbers and letters is called monomial, the sum of several monomials is called polynomial, and monomials and polynomials are collectively called algebraic expressions. ② In a single item, the index sum of all letters is called the number of times of the item. ③ In a polynomial, the degree of the term with the highest degree is called the degree of this polynomial.

Algebraic expression operation: when adding and subtracting, if you encounter brackets, remove them first, and then merge similar items.

Power operation: AM+AN=A(M+N)

(AM)N=AMN

(A/B)N=AN/BN division.

Multiplication of algebraic expressions: ① Multiply the monomial with the monomial, respectively multiply their coefficients and the power of the same letter, and the remaining letters, together with their exponents, remain unchanged as the factors of the product. (2) Multiplying polynomial by monomial means multiplying each term of polynomial by monomial according to the distribution law, and then adding the products. (3) Polynomial multiplied by polynomial. Multiply each term of one polynomial by each term of another polynomial, and then add the products.

There are two formulas: square difference formula/complete square formula.

Algebraic division: ① monomial division, which divides the coefficient and the power of the same base as the factor of quotient respectively; For the letter only contained in the division formula, it is used as the factor of quotient together with its index. (2) Polynomial divided by single item, first divide each item of this polynomial by single item, and then add the obtained quotients.

Factorization: transforming a polynomial into the product of several algebraic expressions. This change is called factorization of this polynomial.

Methods: Common factor method, formula method, grouping decomposition method and cross multiplication were used.

Fraction: ① Algebraic expression A is divided by algebraic expression B. If the divisor B contains a denominator, then this is a fraction. For any fraction, the denominator is not 0. ② The numerator and denominator of the fraction are multiplied or divided by the same algebraic expression that is not equal to 0, and the value of the fraction remains unchanged.

Operation of fraction:

Multiplication: take the product of molecular multiplication as the numerator of the product, and the product of denominator multiplication as the denominator of the product.

Division: dividing by a fraction is equal to multiplying the reciprocal of this fraction.

Addition and subtraction: ① Addition and subtraction with denominator fraction, denominator unchanged, numerator addition and subtraction. ② Fractions with different denominators shall be divided into fractions with the same denominator first, and then added and subtracted.

Fractional equation: ① The equation with unknown number in denominator is called fractional equation. ② The solution whose denominator is 0 is called the root increase of the original equation.

Equations and inequalities

1, equation and equation

Unary linear equation: ① In an equation, there is only one unknown, and the exponent of the unknown is 1. Such an equation is called a one-dimensional linear equation. ② Adding or subtracting or multiplying or dividing (non-0) an algebraic expression on both sides of the equation at the same time, the result is still an equation.

Steps to solve a linear equation with one variable: remove the denominator, shift the term, merge the similar terms, and change the unknown coefficient into 1.

Binary linear equation: An equation that contains two unknowns and all terms are 1 is called binary linear equation.

Binary linear equations: The equations composed of two binary linear equations are called binary linear equations.

A set of unknown values suitable for binary linear equation is called the solution of this binary linear equation.

The common * * * solution of each equation in a binary linear system of equations is called the solution of this binary linear system of equations.

Methods of solving binary linear equations: substitution elimination method/addition and subtraction elimination method.

Quadratic equation of one variable: an equation with only one unknown and the highest coefficient of the unknown term is 2.

1) The relation of quadratic function of quadratic equation in one variable.

Everyone has studied quadratic function (parabola) and has a deep understanding of it, such as solution and representation in images. In fact, the quadratic equation of one variable can also be expressed by quadratic function. In fact, the quadratic equation of one variable is also a special case of quadratic function, that is, when y is 0, it constitutes the quadratic equation of one variable. Then, if expressed in a plane rectangular coordinate system, the quadratic equation of one variable is the intersection of the X axis in the image and the quadratic function. Which is the solution of the equation.

2) Solution of quadratic equation in one variable.

As we all know, a quadratic function has a vertex (-b/2a, 4ac-b2/4a), which is very important. Remember, as mentioned above, the quadratic equation with one variable is also a part of the quadratic function, so he also has his own solution, and he can find all the solutions of the quadratic equation with one variable.

(1) matching method

Using this formula, the equation is transformed into a complete square formula and solved by direct Kaiping method.

(2) Factor decomposition method

Select the common factor, apply the formula, and cross multiply. The same is true for solving quadratic equations with one variable. Using this, the equation can be solved in the form of several products.

(3) Formula method

This method can also be used as a general method to solve quadratic equations with one variable. The roots of the equation are x 1 = {-b+√ [B2-4ac]}/2a, and x2 = {-b-√ [B2-4ac]}/2a.

3) the step of solving a quadratic equation with one variable:

(1) Matching method steps:

First, the constant term is moved to the right of the equation, then the coefficient of the quadratic term is changed to 1, and the square of half the coefficient of 1 is added, and finally the complete square formula is obtained.

(2) The steps of factorization:

Turn the right side of the equation into 0, and then see if you can extract the common factor, formula (here refers to the formula in factorization) or cross multiplication, and if you can, turn it into the form of product.

(3) Formula method

Simply substitute the coefficient of quadratic equation into a variable, where the coefficient of quadratic term is a, the coefficient of linear term is b, and the coefficient of constant term is c.

4) Vieta theorem

Understand with Vieta theorem, Vieta theorem in a quadratic equation, the sum of two roots =-b/a, the product of two roots = c/a.

It can also be expressed as x 1+x2 =-b/a, and x1x2 = c/a. By using Vieta's theorem, we can find out the coefficients in a quadratic equation with one variable, which is very common in the topic.

5) The case of the root of a linear equation with one variable

Using the discriminant of roots to understand, the discriminant of roots can be written as "Delta" and read as "Tune ta", and Delta = B2-4ac can be divided into three situations:

I am △ > 0, a quadratic equation with one variable has two unequal real roots;

II When △=0, the quadratic equation of one variable has two identical real roots;

Three dang △

2. Inequality and unequal groups

Inequality: ① When the symbol > = 0 is used, it passes through quadrant124; When k > 0 and b < 0, pass through quadrant134; When k > 0 and b > 0, pass through quadrant 123. ④ When k > 0, y value increases with the increase of x value, and when x < 0, y value decreases with the increase of x value.

Second, space and graphics.

First, the understanding of graphics

1, point, line, surface

Points, lines and surfaces: ① A figure consists of points, lines and surfaces. (2) Lines intersecting face to face and points where lines intersect. (3) Points become lines, lines become surfaces, and surfaces become bodies.

Unfolding and folding: ① In a prism, the intersection of any two adjacent faces is called an edge, and the side edge is the intersection of two adjacent edges. All sides of the prism are equal in length, the upper and lower bottom surfaces of the prism are the same in shape, and the side surfaces are cuboids. (2) N prism is a prism with N faces on its bottom.

Cutting a geometric figure: cutting a figure with a plane, and the cutting surface is called a section.

Views: main view, left view and top view.

Polygon: It is a closed figure composed of some line segments that are not on the same straight line.

Arc and sector: ① A figure consisting of an arc and two radii passing through the end of the arc is called a sector. ② The circle can be divided into several sectors.

2. Angle

Line: ① A line segment has two endpoints. (2) The line segment extends infinitely in one direction to form a ray. A ray has only one endpoint. ③ A straight line is formed by the infinite extension of both ends of a line segment. A straight line has no end. Only one straight line passes through two points.

Comparison length: ① Of all the connecting lines between two points, the line segment is the shortest. ② The length of the line segment between two points is called the distance between these two points.

Measurement and expression of angle: ① An angle consists of two rays with a common endpoint, and the common endpoint of the two rays is the vertex of the angle. ② One degree of 1/60 is one minute, and one minute of1/60 is one second.

Comparison of angles: ① An angle can also be regarded as a light rotating around its endpoint. (2) The ray rotates around its endpoint. When the ending edge and the starting edge are on a straight line, the angle formed is called a right angle. The starting edge continues to rotate, and when it coincides with the starting edge again, the angle formed is called fillet. (3) The ray from the vertex of an angle divides the angle into two equal angles, and this ray is called the bisector of the angle.

Parallelism: ① Two straight lines that do not intersect in the same plane are called parallel lines. ② One and only one straight line is parallel to this straight line after passing through a point outside the straight line. If both lines are parallel to the third line, then the two lines are parallel to each other.

Perpendicular: ① If two lines intersect at right angles, they are perpendicular to each other. (2) The intersection of two mutually perpendicular straight lines is called vertical foot. ③ On the plane, there is one and only one straight line perpendicular to the known straight line at one point.

Perpendicular bisector: A straight line perpendicular to and bisecting a line segment is called perpendicular bisector.

The perpendicular bisector in perpendicular bisector must be a line segment, not a ray or a straight line, which is related to the infinite extension of rays and straight lines. Look at the back, the middle vertical line is a straight line, so when drawing the middle vertical line, the line segment should pass through two points and then two points (about drawing, we will talk about it later).

Perpendicular bisector theorem;

Property theorem: the distance between the point on the middle vertical line and the two ends of the line segment is equal;

Decision Theorem: The point with the same distance from the endpoint of line segment 2 is on the middle vertical line of this line segment.

Angular bisector: The ray bisecting an angle is called the angular bisector of the angle.

There are several points to note in the definition, that is, the bisector of an angle is a ray, not a line segment or a straight line. Many times there will be a straight line in the topic, which is the symmetry axis of the bisector, which also involves the problem of trajectory. The bisector of an angle is a point with equal distance to both sides of the angle.

Property theorem: the distance between the point on the bisector of an angle and both sides of the angle is equal.

Decision theorem: the points with equal distance to both sides of the angle are on the bisector of the angle.

Square: A set of rectangles with equal adjacent sides is a square.

Properties: Square has all the properties of parallelogram, rhombus and rectangle.

Judgment: 1, rhombus 2 with equal diagonals and rectangle with equal adjacent sides.

Second, the basic theorem

1, there is only one straight line between two points.

2. The line segment between two points is the shortest.

3. The complementary angles of the same angle or equal angle are equal.

4. The complementary angles of the same angle or equal angle are equal.

5. There is one and only one straight line perpendicular to the known straight line.

6. Of all the line segments connecting a point outside the straight line with points on the straight line, the vertical line segment is the shortest.

7. The parallel axiom passes through a point outside the straight line, and there is only one straight line parallel to this straight line.

8. If two straight lines are parallel to the third straight line, the two straight lines are also parallel to each other.

9. The same angle is equal, and two straight lines are parallel.

10, internal dislocation angles are equal, and two straight lines are parallel.

1 1, the inner angles on the same side are complementary, and the two straight lines are parallel.

12, two straight lines are parallel and have the same angle.

13, two straight lines are parallel and the internal dislocation angles are equal.

14. Two straight lines are parallel and complementary.

15, the sum of two sides of a theorem triangle is greater than the third side.

16, the difference between two sides of the inference triangle is smaller than the third side.

17, the sum of the internal angles of the triangle and the theorem triangle is equal to 180.

18, it is inferred that the two acute angles of 1 right triangle are complementary.

19, Inference 2 An outer angle of a triangle is equal to the sum of two inner angles that are not adjacent to it.

20. Inference 3 The outer angle of a triangle is larger than any inner angle that is not adjacent to it.

2 1, the corresponding edge of congruent triangles is equal to the corresponding angle.

22. The edge axiom (SAS) has two edges, and their included angle corresponds to the congruence of two triangles.

23. The corner axiom (ASA) has two corners and two triangles with equal corresponding sides.

24. Inference (AAS) has two angles, and the opposite side of one angle corresponds to the congruence of two triangles.

25. The side-by-side axiom (SSS) has the congruence of two triangles whose three sides correspond to each other.

26. Axiom of hypotenuse and right-angled side (HL) Two right-angled triangles with hypotenuse and a right-angled side are congruent.

27. Theorem 1 The distance from the point on the bisector of the angle to both sides of the angle is equal.

28. Theorem 2 The point where two sides of an angle are equidistant is on the bisector of this angle.

29. The bisector of an angle is the set of all points with equal distance to both sides of the angle.

30, the nature theorem of isosceles triangle The two bottom angles of an isosceles triangle are equal (that is, equilateral angles)

3 1, inference 1 The bisector of the vertex of the isosceles triangle bisects the base and is perpendicular to the base.

32. The bisector of the top angle, the median line on the bottom edge and the height on the bottom edge of the isosceles triangle coincide with each other.

33. Inference 3 All angles of an equilateral triangle are equal, and each angle is equal to 60.

34. Decision theorem of isosceles triangle If a triangle has two equal angles, then the sides of the two angles are also equal (equal angles and equal sides).

35. Inference 1 A triangle with three equal angles is an equilateral triangle.

Inference 2 An isosceles triangle with an angle equal to 60 is an equilateral triangle.

37. In a right triangle, if an acute angle is equal to 30, then the right side it faces is equal to half of the hypotenuse.

38. The midline of the hypotenuse of a right triangle is equal to half of the hypotenuse.

39. Theorem The point on the vertical line of a line segment is equal to the distance between the two endpoints of this line segment.

40. The inverse theorem and the equidistant point between the two endpoints of a line segment are on the vertical line of this line segment.

4 1, the middle vertical line of a line segment can be regarded as the set of all points with equal distance at both ends of the line segment.

42. Theorem 1 Two graphs symmetric about a straight line are conformal.

43. Theorem 2 If two figures are symmetrical about a straight line, then the symmetry axis is the perpendicular line connecting the corresponding points.

44. Theorem 3 Two figures are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection point is on the axis of symmetry.

45. Inverse Theorem If the straight line connecting the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.

46. Pythagorean Theorem The sum of squares of two right-angled sides A and B of a right-angled triangle is equal to the square of hypotenuse C, that is, a2+b2=c2.

47. Inverse Theorem of Pythagorean Theorem If the three sides of a triangle A, B and C are related to a2+b2=c2, then this triangle is a right triangle.

48. The sum of the internal angles of a quadrilateral is equal to 360 degrees.

49. The sum of the external angles of the quadrilateral is equal to 360.

50. Theorem The sum of the interior angles of a polygon is equal to (n-2) × 180.

5 1, it is inferred that the sum of the external angles of any polygon is equal to 360.

52. parallelogram property theorem 1 parallelogram diagonal is equal

53. parallelogram property theorem 2 The opposite sides of a parallelogram are equal

54. It is inferred that the parallel segments sandwiched between two parallel lines are equal.

55, parallelogram property theorem 3 diagonal bisection of parallelogram.

56. parallelogram judgment theorem 1 Two groups of quadrilaterals with equal diagonals are parallelograms.

57. parallelogram decision theorem 2 Two groups of quadrilaterals with equal opposite sides are parallelograms.

58. parallelogram decision theorem 3 The quadrilateral whose diagonals are bisected is a parallelogram.

59. parallelogram decision theorem 4 A set of parallelograms with equal opposite sides is a parallelogram.

60. Theorem of Rectangular Properties 1 All four corners of a rectangle are right angles.

6 1, rectangle property theorem 2 The diagonals of rectangles are equal.

62. Rectangular Decision Theorem 1 A quadrilateral with three right angles is a rectangle.

63. Rectangular Decision Theorem 2 A parallelogram with equal diagonals is a rectangle.

64. Diamond Property Theorem 1 All four sides of a diamond are equal

65. Diamond Property Theorem 2 Diagonal lines of diamonds are perpendicular to each other, and each diagonal line bisects a set of diagonal lines.

66, diamond area = half of the diagonal product, that is, S=(a×b)÷2.

67. Diamond Decision Theorem 1 A quadrilateral with four equilateral sides is a diamond.

68. Diamond Decision Theorem 2 Parallelograms with diagonal lines perpendicular to each other are diamonds.

69. Theorem of Square Properties 1 Four corners of a square are right angles and four sides are equal.

70. Theorem of Square Properties 2 The two diagonals of a square are equal and bisected vertically, and each diagonal bisects a set of diagonals.

7 1 and theorem 1 are congruent for two centrally symmetric graphs.

72. Theorem 2 For two graphs with symmetric centers, the connecting lines of symmetric points pass through the symmetric centers and are equally divided by the symmetric centers.

73. Inverse Theorem If a straight line connecting the corresponding points of two graphs passes through a certain point and is equally divided by the point, then the two graphs are symmetrical about the point.

74, isosceles trapezoid property theorem isosceles trapezoid on the same bottom of the two angles are equal.

75. The two diagonals of an isosceles trapezoid are equal.

76. Isosceles Trapezoids Decision Theorem Two isosceles trapeziums on the same base are isosceles trapeziums.

77. A trapezoid with equal diagonal lines is an isosceles trapezoid.

78. Theorem of Equal Segment of Parallel Lines If a group of parallel lines have the same segment on a straight line, then the segments on other straight lines are the same.

79. Inference 1 passes through a straight line parallel to the trapezoid waist bottom, and the other waist will be equally divided.

80. Inference 2 A straight line passing through the midpoint of one side of a triangle and parallel to the other side will bisect the third side.

8 1, the midline theorem of a triangle The midline of a triangle is parallel to the third side and equal to half of it.

82. The trapezoid midline theorem is parallel to the two bottoms and equal to half of the sum of the two bottoms L = (a+b) ÷ 2s = l× h.

83. The basic property of (1) ratio: If a:b=c:d, then ad=bc If ad=bc, then A: B = C: D.

84.(2) Combinatorial properties: If A/B = C/D, then (A B)/B = (C D)/D.

85.(3) Isometric property: If A/B = C/D = … = M/N (B+D+…+N ≠ 0),

Then (a+c+…+m)/(b+d+…+n) = a/b.

86. Proportional theorem of parallel line segments Three parallel lines cut two straight lines, and the corresponding line segments are proportional.

87. It is inferred that a straight line parallel to one side of a triangle cuts the other two sides (or extension lines on both sides), and the corresponding line segments obtained are proportional.

Theorem If the corresponding line segments obtained by cutting two sides (or extension lines of two sides) of a triangle are proportional, then this straight line is parallel to the third side of the triangle.

89. A straight line parallel to one side of a triangle and intersecting with the other two sides, the three sides of the triangle are proportional to the three sides of the original triangle.

Theorem A straight line parallel to one side of a triangle intersects the other two sides (or extension lines of both sides), and the triangle formed is similar to the original triangle.

9 1, similar triangles's decision theorem 1 Two angles are equal and two triangles are similar (ASA)

92. Two right-angled triangles divided by the height on the hypotenuse are similar to the original triangle.

93. Decision Theorem 2: Two sides are proportional and the included angle is equal, and two triangles are similar (SAS).

94. Decision Theorem 3 Three sides are proportional and two triangles are similar (SSS).

Theorem If the hypotenuse and a right-angled side of a right-angled triangle are proportional to the hypotenuse and a right-angled side of another right-angled triangle, then the two right-angled triangles are similar.

96. The property theorem 1 similar triangles corresponds to the height ratio, the ratio of the corresponding centerline and the ratio of the corresponding angular bisector are all equal to the similarity ratio.

97. Property Theorem 2 The ratio of similar triangles perimeter is equal to the similarity ratio.

98. Property Theorem 3 The ratio of similar triangles area is equal to the square of similarity ratio.

99. The sine value of any acute angle is equal to the cosine value of the remaining angles, and the cosine value of any acute angle is equal to the sine value of the remaining angles.

100, the tangent of any acute angle is equal to the cotangent of other angles, and the cotangent of any acute angle is equal to the tangent of other angles.

10 1. 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.

103, the outside of the circle can be regarded as a collection of points whose center distance is greater than the radius.

104, same circle or same circle radius.

105. The trajectory of a point whose distance to a fixed point is equal to a fixed length is a circle with the fixed point as the center and the fixed length as the radius.

106, it is known that the locus of the point where the two endpoints of a line segment are equidistant is the midline of the line segment.

107, it is known that the locus of points with equal distance on both sides of an angle is the bisector of this angle.

108, the locus to the equidistant points of two parallel lines is a straight line parallel to and equidistant from these two parallel lines.

109. Theorem Three points that are not on the same straight line determine a circle.

1 10, the vertical diameter theorem bisects the chord perpendicular to the diameter of the chord and bisects the two arcs opposite the chord.

1 1 1, reasoning 1

(1) bisects the diameter (not the diameter) of the chord perpendicular to the chord and bisects 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.

1 12, it is inferred that the arcs between two parallel chords of a circle are equal.

1 13. A circle is a centrally symmetric figure with the center of the circle as the center of symmetry.

1 14. Theorem In the same circle or in the same circle, the isocentric angle has equal arc, chord and chord center distance.

1 15. It is inferred that in the same circle or the same circle, if one set of quantities in two central angles, two arcs, two chords or the distance between two chords is equal, the corresponding other set of quantities is also equal.

1 16, Theorem The angle of an arc is equal to half its central angle.

1 17, it is inferred that 1 the circumferential angles of the same arc or equivalent arc are equal; In the same circle or in the same circle, the arcs of equal circumferential angles are also equal.

1 18, it is inferred that the circumferential angle (or diameter) of the semicircle is a right angle; A chord with a circumferential angle of 90 is a diameter.

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 a right triangle.

120, it is proved that the diagonals of the inscribed quadrilateral of a circle are complementary, and any external angle is equal to its internal diagonal.

12 1, ① the 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.

122, tangent judgment theorem The straight line passing through the outer end of the radius and perpendicular to this radius is the tangent of the circle.

123, the property theorem of tangent. The tangent of a circle is perpendicular to the radius passing through the tangent point.

124, inference 1 A straight line passing through the center of the circle and perpendicular to the tangent must pass through the tangent point.

125, inference 2 A straight line passing through the tangent and perpendicular to the tangent must pass through the center of the circle.

126. The tangent length theorem leads to two tangents of the circle from a point outside the circle. Their tangents have the same center, and the connecting line of this point bisects the included angle of the two tangents.

127, the sum of two opposite sides of the circumscribed quadrangle of a circle is equal.

128, chord angle theorem chord angle is equal to the circumferential angle of the arc pair it clamps.

129. From this, it can be inferred that if the arc sandwiched between two chordal angles is equal, then the two chordal angles are also equal.

130, intersection chord theorem The product of the length of two intersecting chords divided by the intersection point in a circle is equal.

13 1. It is inferred that if the chord intersects the diameter vertically, then half of the chord is the proportional average of the two line segments formed by its divided diameter.

132, the tangent theorem leads to the tangent and secant of the circle from a point outside the circle, and the tangent length is the median term of the ratio of the lengths of the two lines from this point to the intersection of the secant and the circle.

133. 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.

134, if two circles are tangent, then the tangent point must be on the line.