According to the internal relationship between the conditions and conclusions of mathematical problems, this paper not only analyzes its algebraic significance, but also reveals its geometric significance. Make the relationship between quantity and figure skillfully and harmoniously combined, and make full use of this combination to seek ideas and solve problems.

2. The concept of connection and transformation

Things are interrelated, restricted and transformed. All parts of mathematics are also interrelated and can be transformed into each other.

When solving problems, if we can properly handle the mutual transformation between them, we can often turn the difficult into the easy and simplify the complicated.

Such as: substitution transformation, known and unknown transformation, special and general transformation, concrete and abstract transformation, partial and whole transformation, dynamic and static transformation and so on.

3. The idea of classified discussion

In mathematics, we often need to investigate the research object under different circumstances according to its different nature; This classified thinking method is an important mathematical thinking method and an important problem-solving strategy.

4. undetermined coefficient method

When the mathematical formula we study has a certain form, we only need to find the value of the letter to be determined in the formula to determine it. Therefore, substituting the known conditions into the formula with undetermined form will often produce an equation or equation group with undetermined letters, and then solving this equation or equation group can solve the problem.

5. Matching method

It is to construct an algebraic expression in a flat way and then make the necessary modifications. Matching method is an important deformation skill in junior high school algebra, which plays an important role in decomposing factors, solving equations and discussing quadratic functions.

6. Alternative methods

In the process of solving problems, in order to solve problems further, the formula of one or several letters is taken as a whole and represented by a new letter. Method of substitution can simplify a complicated formula and turn the problem into a more basic problem than the original one, so as to achieve the purpose of simplifying the complex and turning the difficult into the easy.

7. Analytical method

When studying or proving a proposition, it is not obvious to trace back to the known conditions from the conclusion and deduce the sufficient conditions for its establishment from the conclusion; Then take this as a conclusion, and further study the sufficient conditions for its establishment until it reaches the known conditions, thus proving the proposition. This kind of thinking process is often called "grasping the fruit and finding the cause"

8. Integrated approach

When studying or proving a proposition, if the direction of reasoning is to proceed from known conditions and draw conclusions step by step, this thinking process is usually called "from cause to effect"

9. Deductive method

Reasoning method from general to special.

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Reasoning method from general to special.

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Among many objective things, there are some things with similar nature, which are between two or two things; According to the fact that some of their attributes are the same or similar, the reasoning method that they may be the same or similar in other attributes is deduced. Analogy can be special to special, or general to general reasoning.

Functions, equations, inequalities

Common mathematical thinking methods:

(1) The thinking method of combining numbers and shapes.

⑵ undetermined coefficient method.

(3) Matching method.

(4) the idea of connection and transformation.

5. Translation and transformation of images.

Prove that the angles are equal

1. The vertex angles are equal.

2. The complementary angle or complementary angle of an angle (or the same angle) is equal.

3. Two straight lines are parallel, with the same angle and the same internal dislocation angle.

4. All right angles are equal.

5. The two angles divided by the bisector are equal.

6. In the same triangle, equal sides are equal angles.

7. In an isosceles triangle, the height (or midline) of the base bisects the vertex.

8. The diagonals of parallelograms are equal.

9. Each diagonal of the diamond bisects a set of diagonal lines.

10. The isosceles trapezoid is equal to the two angles on the base.

1 1. Relation Theorem: If two arcs (or chords, or chord distances) are equal in the same circle or equal circle, then the central angles they face are equal.

12. Any external angle of a quadrilateral inscribed with a circle is equal to its internal angle.

13. The circumferential angles of the same arc or equal arc are equal.

14. The chord tangent angle is equal to the circumferential angle of the arc pair it clamps.

15. In the same circle or in the same circle, if the arcs sandwiched between the two tangent angles are equal, then the two tangent angles are also equal.

16. The corresponding angles of congruent triangles are equal.

17. The corresponding angles of similar triangles are equal.

18. Use equivalent substitution.

19. Calculate the angle equality by algebra and trigonometry.

20. Tangent Length Theorem: The two tangents of a circle drawn from a point outside the circle are equal in length, and the connecting line between the point and the center of the circle bisects the included angle of the two tangents.

Prove the parallelism or verticality of a straight line

1. The main basis and method to prove that two lines are parallel.

(1) Two disjoint lines defined on the same plane are parallel.

(2) Parallelism theorem, two straight lines are parallel to the third straight line, and these two straight lines are also parallel to each other.

⑶ Determination of parallel lines: The same complementary angle is equal (internal angle or internal angle of the same side), and the two straight lines are parallel.

(4) The opposite sides of the parallelogram are parallel.

5] The two bottoms of the trapezoid are parallel.

[6] The midline of a triangle (or trapezoid) is parallel to the third side (or two bottom sides).

(7) If the corresponding line segments cut by two sides (or extension lines of two sides) of a triangle are proportional, the straight line is parallel to the third side of the triangle.

2. The main basis and method to prove that two straight lines are perpendicular.

(1) When one of the four angles formed by the intersection of two straight lines is a right angle, the two straight lines are perpendicular to each other.

(2) The two right angles of a right triangle are perpendicular to each other.

(3) If the two acute angles of a triangle are complementary, the third inner angle is a right angle.

(4) If the median line of one side of a triangle is equal to half of this side, the triangle is a right triangle.

[5] The square of one side of a triangle is equal to the sum of the squares of the other two sides, so the inner angle subtended by this side is a right angle.

The height of one side of a triangle (or polygon) is perpendicular to this side.

(7) The bisector of the top angle of an isosceles triangle (or the median line on the bottom) is perpendicular to the bottom.

The two sides of a rectangle are perpendicular to each other.

Diagonal lines of diamonds are perpendicular to each other.

⑽ The diameter (non-diameter) of the bisector is perpendicular to this chord, or the diameter of the arc subtended by the bisector is perpendicular to this chord.

⑾ The circumferential angle of a semicircle or diameter is a right angle.

⑿ The tangent of the circle is perpendicular to the radius of the tangent point.

[13] The intersection of two circles is perpendicular to the common chord of the two circles.