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 the same 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, (1) the basic nature of the ratio:
If a:b=c:d, then ad=bc.
If ad=bc, then a: b = c: d.
84, (2) the ratio nature:
If a/b = c/d, then (a b)/b = (c d)/d.
85, (3) equal ratio properties:
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.
135, ① the distance between two circles is d > r+r+r.
(2) circumscribed circle d 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).
136, theorem The intersection line of two circles bisects the common chord of two circles vertically.
137, the theorem divides the 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 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
139 and each inner angle of a regular n-polygon is equal to (n-2) ×180/n.
140, theorem The radius of a regular N-polygon and apothem divide the regular N-polygon into 2n congruent right triangles.
14 1, and the area of the regular n-polygon Sn = PNRN/2 p represents the perimeter of the regular n-polygon.
142, and the regular triangle area √ 3a/4a indicates the side length.
143. if there are k positive n corners around a vertex, since the sum of these corners should be 360, then K× (n-2) 180/n = 360 becomes (n-2)(k-2)=4.
144. calculation formula of arc length: L = NR/ 180.
145, sector area formula: s sector =n r 2/360 = lr/2.
146, inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)
Sine theorem a/sinA=b/sinB=c/sinC=2R.
Note: where r represents the radius of the circumscribed circle of the triangle.
Cosine theorem b2=a2+c2-2accosB
Note: Angle B is the included angle between side A and side C..
Fourth, the basic method
1, matching method
The so-called formula is to change some items of an analytical formula into the sum of positive integer powers of one or more polynomials by using the method of constant deformation. The method of solving mathematical problems with formulas is called matching method. Among them, the most common method is to make it completely flat. Matching method is an important method of constant deformation in mathematics. It is widely used in factorization, simplifying roots, solving equations, proving equality and inequality, finding extreme values of functions and analytical expressions.
2, factorization method
Factorization is to transform a polynomial into the product of several algebraic expressions. Factorization is the basis of identity deformation. As a powerful mathematical tool and method, it plays an important role in solving algebra, geometry and trigonometry problems. There are many methods of factorization, such as extracting common factors, formulas, grouping decomposition, cross multiplication and so on. Middle school textbooks also introduce the use of decomposition and addition, root decomposition, exchange elements, undetermined coefficients and so on.
3. Alternative methods
Method of substitution is a very important and widely used method to solve problems in mathematics. We usually refer to unknowns or variables as elements. The so-called method of substitution is to replace a part of the original formula with new variables in a complicated mathematical formula, thus simplifying it and making the problem easy to solve.
4. Discriminant method and Vieta theorem.
The root discrimination of unary quadratic equation ax2+bx+c=0(a, B, C belongs to R, a≠0) and△ = B2-4ac is not only used to judge the properties of roots, but also widely used in algebraic deformation, solving equations (groups), solving inequalities, studying functions and even geometric and trigonometric operations as a problem-solving method.
Vieta's theorem not only knows one root of a quadratic equation, but also finds another root. Knowing the sum and product of two numbers, we can find the symmetric function of the root, calculate the sign of the root of quadratic equation, solve the symmetric equation and solve some problems about quadratic curve. , has a very wide range of applications.
5, undetermined coefficient method
When solving mathematical problems, it is first judged that the obtained results have a certain form, which contains some undetermined coefficients, then the equations about undetermined coefficients are listed according to the problem setting conditions, and finally the values of these undetermined coefficients or some relationship between these undetermined coefficients are found. This method is called undetermined coefficient method to solve mathematical problems. It is one of the commonly used methods in middle school mathematics.
6. Construction method
When solving problems, we often use this method to construct auxiliary elements by analyzing conditions and conclusions, which can be a figure, an equation (group), an equation, a function, an equivalent proposition and so on. And establish a bridge connecting conditions and conclusions, so that the problem can be solved. This mathematical method of solving problems is called construction method. Using construction method to solve problems can make algebra, trigonometry, geometry and other mathematical knowledge permeate each other, which is beneficial to solving problems.
7. reduce to absurdity
Reduction to absurdity is an indirect proof method. First, a hypothesis contrary to the conclusion of the proposition is put forward, and then from this hypothesis, through correct reasoning, contradictions are led out, thus denying the opposite hypothesis and affirming the correctness of the original proposition. The reduction to absurdity can be divided into reduction to absurdity (with only one opposite conclusion) and exhaustive reduction to absurdity (with more than one opposite conclusion). The steps to prove a proposition by reduction to absurdity can be roughly divided into: (1) reverse design; (2) return to absurdity; (3) conclusion.
Counterhypothesis is the basis of reduction to absurdity. In order to make a correct counter-hypothesis, we need to master some commonly used negative expressions, such as yes and no; To exist or not to exist; Parallel or nonparallel; Vertical or not vertical; Equal to or not equal to; Big (small), not big (small); Both, not all; At least one, no; At least n, at most (n-1); At most one, at least two; Only, at least two.
Reduction to absurdity is the key to reduction to absurdity. There is no fixed model in the process of derivation of contradiction, but it must be based on reverse design, otherwise the derivation will become passive water and trees without roots. Reasoning must be rigorous. There are the following types of contradictions: contradictions with known conditions; Contradicting with known axioms, definitions, theorems and formulas; There are dual contradictions; Contradictions
8. Find the area method
The area formula in plane geometry and the property theorems related to area calculation derived from the area formula can be used not only to calculate the area, but also to prove that plane geometry problems sometimes get twice the result with half the effort. The method of proving or calculating plane geometric problems by using area relation is called area method, which is commonly used in geometry.
The difficulty in proving plane geometry problems by induction or analysis lies in adding auxiliary lines. The characteristic of area method is to connect the known quantity with the unknown quantity by area formula, and achieve the verification result through operation. Therefore, using the area method to solve geometric problems, the relationship between geometric elements becomes the relationship between quantities, and only calculation is needed. Sometimes there may be no auxiliary lines, even if auxiliary lines are needed, it is easy to consider.
9, geometric transformation method
In the study of mathematical problems, the transformation method is often used to transform complex problems into simple problems and solve them. The so-called transformation is a one-to-one mapping between any element of a set and the elements of the same set. The transformation involved in middle school mathematics is mainly elementary transformation. There are some exercises that seem difficult or even impossible to start with. We can use geometric transformation to simplify the complex and turn the difficult into the easy. On the other hand, the transformed point of view can also penetrate into middle school mathematics teaching. It is helpful to understand the essence of graphics by combining the research of graphics under isostatic conditions with the research of motion.
Geometric transformation includes: (1) translation; (2) rotation; (3) symmetry.
10, objective problem solving method
Multiple-choice questions are questions that give conditions and conclusions and require finding the correct answer according to a certain relationship. Multiple-choice questions are ingenious in conception and flexible in form, which can comprehensively examine students' basic knowledge and skills, thus increasing the capacity and knowledge coverage of test papers.
Fill-in-the-blank question is one of the important questions in standardized examination. Like multiple-choice questions, it has the advantages of clear test objectives, wide knowledge coverage, accurate and fast marking, and is conducive to examining students' analytical judgment and calculation ability. The difference is that the fill-in-the-blank question does not give an answer, which can prevent students from guessing the answer.
In order to solve multiple-choice questions and fill-in-the-blank questions quickly and correctly, in addition to accurate calculation and strict reasoning, there are also methods and skills to solve multiple-choice questions and fill-in-the-blank questions. The following examples introduce common methods.
(1) Direct deduction method: Starting directly from the conditions given by the proposition, using concepts, formulas, theorems, etc. Carry out reasoning or operation, draw a conclusion and choose the correct answer. This is the traditional method of solving problems, which is called direct deduction.
(2) Verification method: find out the appropriate verification conditions from the questions, and then find out the correct answer through verification, or substitute alternative answers into the conditions for verification to find out the correct answer. This method is called verification method (also called substitution method). This method is often used when encountering quantitative propositions.
(3) Special element method: substitute appropriate special elements (such as figures or numbers) into the conditions or conclusions of the topic, so as to get the solution. This method is called the special element method.
(4) Exclusion and screening method: for multiple-choice questions with only one correct answer, according to mathematical knowledge or reasoning and calculus, the incorrect conclusion is excluded and the remaining conclusions are screened, so that the solution to make the correct conclusion is called exclusion and screening method.
(5) Graphic method: The method of judging and making a correct choice through the properties and characteristics of the graphics or images that meet the conditions of the topic is called graphic method. Graphic method is one of the common methods to solve multiple-choice questions.
(6) Analysis method: directly through the conditions and conclusions of multiple-choice questions, make detailed analysis, induction and judgment, so as to select the correct result, which is called analysis method.
People say that geometry is difficult, and it is difficult in auxiliary lines.
Auxiliary line, how to add it? Master theorems and concepts.
We must study hard and find out the rules by experience.
There is an angular bisector in the picture, which can be perpendicular to both sides.
You can also look at the picture in half, and there will be a relationship after symmetry.
Angle bisector parallel lines, isosceles triangles add up.
Angle bisector plus vertical line, try three lines.
Perpendicular bisector is a line segment that usually connects the two ends of a straight line.
It needs to be proved that the line segment is double-half, and extension and shortening can be tested.
The two midpoints of a triangle are connected to form a midline.
A triangle has a midline and the midline extends.
A parallelogram appears and the center of symmetry bisects the point.
Make a high line in the trapezoid and try to translate a waist.
It is common to move diagonal lines in parallel and form triangles.
The card is almost the same, parallel to the line segment, adding lines, which is a habit.
In the proportional conversion of equal product formula, it is very important to find the line segment.
Direct proof is more difficult, and equivalent substitution is less troublesome.
Make a high line above the hypotenuse, which is larger than the middle term.
Calculation of radius and chord length, the distance from the chord center to the intermediate station.
If there are all lines on the circle, the radius of the center of the tangent point is connected.
Pythagorean theorem is the most convenient for the calculation of tangent length.
To prove that it is tangent, carefully distinguish the radius perpendicular.
Is the diameter, in a semicircle, to connect the chords at right angles.
An arc has a midpoint and a center, and the vertical diameter theorem should be remembered completely.
There are two chords on the corner of the circle, and the diameters of the two ends of the chords are connected.
Find tangent chord, same arc diagonal, etc.
If you want to draw a circumscribed circle, draw a vertical line in the middle on both sides.
Also make an inscribed circle, and the bisector of the inner corner is a dream circle.
If you meet an intersecting circle, don't forget to make it into a string.
Two circles tangent inside and outside pass through the common tangent of the tangent point.
If you add a connector, the tangent point must be on the connector.
Adding a circle to the equilateral angle makes it not so difficult to prove the problem.
The auxiliary line is a dotted line, so be careful not to change it when drawing.
If the graph is dispersed, rotate symmetrically to carry out the experiment.
Basic drawing is very important and should be mastered skillfully.
You should pay more attention to solving problems and often sum up the methods clearly.
Don't blindly add lines, the method should be flexible.
No matter how difficult it is to choose the analysis and synthesis methods, it will be reduced.
Study hard and practice hard with an open mind, and your grades will soar.