A triangle is one of the polygons with the least number of sides. Its definition is: a figure composed of three line segments that are not on the same line and are connected end to end is called a triangle.

The condition that three line segments are not on a straight line. If three line segments are on a straight line, we think that a triangle does not exist. The other three line segments must be connected end to end, which means that the triangle figure must be closed. A triangle has three sides, three angles and three vertices.

The main part of a triangle.

The main line segments in a triangle are: bisector, median line and height line of the triangle.

These three lines must be mastered by drawing on the basis of understanding and mastering their definitions. For these three line segments, three points must be made clear:

(1) The bisector, median line and height line of a triangle are all line segments, not straight lines or rays.

(2) A triangle has three bisectors, a median line and a height line, all of which are inside the triangle. When △ABC is an acute triangle, the three heights of the triangle are all inside the triangle, the two heights of an obtuse triangle fall on the extension lines of the sides, and the two heights are outside the triangle. The two heights of a right triangle are just its two right-angled sides.

(3) When drawing the bisector, midline and height of three angles of a triangle, we can find that they all intersect at one point. We can give concrete proof later. In the future, we will call the intersection of bisectors of three angles of a triangle the center of the triangle, the intersection of three midlines the center of gravity of the triangle, and the intersection of three heights the vertical center of the triangle.

Edge classification of triangle

Some of the three sides of a triangle are unequal, some are equal and some are all equal. Therefore, triangles are classified according to the equal relationship of sides as follows:

An equilateral triangle is a special case of an isosceles triangle.

The basis for judging whether three sides can form a triangle

The three sides of △ABC are A, B and C respectively. According to the axiom, "among all the straight lines connecting two points, the line segment is the shortest". Understand:

③a+b>c，①a+c>b，②b+c>a

Theorem: The sum of any two sides of a triangle is greater than the third side.

B-a < c and b-a >-c are obtained from ② and ③.

So | a-b | < c, in the same way | b-c | < a, | a-c | < b.

It can be inferred from this that:

The difference between any two sides of a triangle is less than the third side.

The above theorems and inferences are actually two ways of describing a problem. Theorem contains inference, and inference can also replace theorem. In addition, theorems and inferences are the basis for judging whether three line segments can form a triangle. For example, if the lengths of three line segments are 5, 4 and 3 respectively, a triangle can be formed, but if the lengths of three line segments are 5, 3 and 1 respectively, a triangle cannot be formed.

Judge whether three sides can form a triangle.

For a certain side, such as side A, as long as | b-c | < a < b+c is satisfied, a triangle can be formed. This is because | b-c | < a, that is, b-c < a, and b-c > b-c and a+b > c, plus b+c > a, satisfy the condition that the sum of any two sides is greater than the third side. On the other hand, as long as the three line segments A, B and C meet the conditions of forming a triangle, there must be | B-C | < A < B+C.

Under special circumstances, if the line segment A is known to be the largest, as long as B+C >; A can determine that three line segments A, B and C can form a triangle. At the same time, if the line segment A is known to be the smallest, as long as | b-c | < a is satisfied, it can be judged that the three line segments A, B and C form a triangle.

Proof of the theorem of sum of interior angles of triangle

In addition to the proof methods given in the textbook, there are many kinds of proof methods. Here, I will introduce the ideas of two proof methods:

As shown in the figure, the method 1 passes through vertex a in the form of DE‖BC,

Using the properties of parallel lines, we can get ∠ b = ∠B=∠2,

∠ c =∠ 1, thus proving the interior angle of the triangle.

The sum equals a right angle ∠DAE.

Method 2, as shown in the figure, takes the BC side of △ABC.

A little d, more than d is DE‖AB, DF‖AC,

Pay AC and AB to E and F respectively, and then use parallelism.

The properties of straight lines can prove that the sum of the internal angles of △ABC is equal to.

Boxer ∠BDC。

Triangles are classified by angle.

According to the triangle interior angle theorem, any interior angle of a triangle is less than 180, and its interior angle may be acute, right or obtuse.

Triangles can be classified by angle as follows:

According to the triangle interior angle sum theorem, the following inferences can be made:

It is inferred that the two acute angles of 1 right triangle are complementary.

Inference 2 An outer angle of a triangle is equal to the sum of two non-adjacent inner angles.

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

At the same time, we can easily draw the following conclusions:

(1) A triangle has at most one right angle or obtuse angle.

(2) At least two internal angles of a triangle are acute angles.

(3) At least one angle of the triangle is equal to or less than 60 (otherwise, if all three internal angles are greater than 60; Then the sum of the internal angles of this triangle is greater than 180, which contradicts the theorem).

(4) A triangle has six external angles, every two of which are equal, so the sum of the three external angles of the triangle is equal to 360.

The nature of congruent triangles

Two Basic Properties of congruent triangles

The edges corresponding to (1) congruent triangles are equal.

(2) The corresponding angles of congruent triangles are equal.

Determine the corresponding edges and angles of two congruent triangles.

How to find the corresponding edges and angles of two congruent triangles accurately and quickly according to the known conditions? The method can be summarized as follows:

(1) If two angles are equal, it is a corresponding angle, and the opposite side of the corresponding angle is a corresponding edge.

(2) If two sides are equal, these two sides are corresponding sides, and the opposite corners of the corresponding sides are corresponding angles.

(3) The edge sandwiched by two corresponding corners is the corresponding edge.

(4) The included angle between two corresponding edges is the corresponding angle.

Judging the congruence of triangles from congruent triangles's definition

According to congruent triangles's definition, to judge whether two triangles are congruent, it is necessary to know whether three sides and three angles are equal. However, in application, it is very troublesome to judge whether two triangles are congruent by definition, so it is necessary to find the conditions that can completely judge a triangle, so as to judge whether two triangles are congruent with fewer conditions and simple methods.

Axiom for judging the coincidence of sides, angles and sides of two triangles

Content: There are two congruent triangles with two sides and their included angles are equal.

This judgment method is given in the form of axioms, and we can verify it through practical operation, but verification does not mean proof, so we should distinguish it.

The condition of axiom is three elements: edge, angle and edge, which means that two edges and the angle between them are equal. It cannot be understood that two sides are equal to one of the angles. Otherwise, these two triangles are not necessarily congruent.

For example, in △ABC and △ a ′ b ′ c ′,

As shown in the right figure, AB=A'B', ∠A=∠A',

BC=A'C', but △ABC is not all equal.

△A′B′C ′.

For another example, in the right figure, in △ABC and △ A ′ B ′ C ′, AB = A ′ B ′, ∠ B = ∠ B ′, AC = A ′ C ′, but △ABC and △ A ′ B ′ C ′ are incomplete.

The reason is that two sides and an angle are equal, isn't it?

Two sides required by axiom and the clip of these two sides

Angle corresponds to the condition of equality.

Note: As can be seen from the above two examples, SAS≠SSA.

The second axiom for judging the coincidence of two triangles

Content: Two triangles with two angles and their corresponding edge congruence (ASA).

This axiom should be further understood through drawing and experiment.

Axiom emphasizes that two angles and their sides are equal and corresponding, which essentially contains a sequential relationship. It must not be understood that in a triangle, it is two angles and their sides, while in another triangle, it is the opposite side of two angles and one of them.

As shown on the right, in △ABC and △ A ′ B ′ C ′,

∠A =∠A′，∠B =∠B′，AB = A′C′，

But these two triangles are obviously not equal. as a result of

I didn't pay attention to the word "correspondence" in the axiom.

The order of edges, angles and edges in axiom 1 cannot be changed, that is, SAS cannot be changed to SSA or ASS. And Asa.

Axiom can change its order, it can be changed to AAS or SAA, but the word "correspondence" between two triangles cannot be changed. At the same time, this axiom reflects that two angles are equal, which is essentially that three angles in two triangles are equal, so we only need to pay attention to one corresponding edge in the application process.

According to axiom 2, a right triangle with two acute angles corresponding to one side is congruent.

Axioms for Determining Equivalence of Two Triangles: Edge, Edge and Edge

Axiom: three sides correspond to the congruence of two triangles (that is, edge, edge axiom).

When the axiom of edge, edge and edge judges the congruence of two triangles, the corresponding edges are equal.

This axiom tells us that as long as the lengths of the three sides of a triangle are determined, the shape of the triangle is completely determined. This is the stability of the triangle.

Determine the consistency of two triangles

Through the study of the above three axioms, we can know that when judging the congruence of two triangles, it is not necessary to judge that the three sides of a triangle and two triangles are equivalent according to the definition, only three pairs of conditions are needed.

Any three combinations of the six conditions of a triangle. There are only the following situations:

(1) Three sides are equal.

(2) Two sides and an angle are equal.

(3) Two corners on one side are equal.

(4) Triangle correspondence is equal.

HL axiom

We know that two triangles that satisfy the equivalence of edges, edges and angles are not necessarily identical.

However, for two right triangles, this conclusion must be true.

Axiom of hypotenuse and right-angled edge: Two right-angled triangles have hypotenuse and a right-angled edge (abbreviated as HL).

The proposition of this axiom is essentially that the three elements are equal, which itself contains a rectangular equality. The core of two triangles with equal sides and angles is the condition that an angle is a right angle. Because the right triangle is a special triangle, the four judgment methods learned in the past are applicable to the right triangle as usual.

Property theorem and inverse theorem of angular bisector

Property theorem: the distance from a point on the bisector of an angle to both sides of this angle is equal.

Inverse theorem: equidistant points on both sides of an angle are on the bisector of this angle.

A point on the bisector of an angle is equal to the distance on both sides of the angle.

The property theorem and inverse theorem of angular bisector expressed in symbolic language

Attribute theorem:

P is on the bisector of ∠AOB.

PD⊥OA,PE⊥OB

∴PD=PE

Inverse theorem:

∵PD=PE,PD⊥OA,PE⊥OB

Point p is on the bisector of ∠AOB.

Definition of angular bisector

If a ray divides an angle into two equal angles, then this ray is called the bisector of this angle.

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

Properties of bisector of triangle angle

The three bisectors of a triangle intersect at a point, and the distances from the intersection to the three sides are equal.

Reciprocal proposition

In two propositions, if the topic of the first proposition is the conclusion of the second proposition, and the conclusion of the first proposition is the topic of the second proposition, then these two propositions are called reciprocal propositions. If one of them is called the original proposition, then the other is called its inverse proposition.

Authenticity of Original Proposition and Inverse Proposition

Every proposition has an inverse proposition, but the original proposition is true, and its inverse proposition is not necessarily true. There are four kinds of truth and falsehood in the original proposition and the inverse proposition: truth and falsehood; True, true; Fake, fake; Virtual and real are born together.

Reciprocity theorem

If the inverse proposition of a theorem is proved to be true, it is also a theorem. These two theorems are called reciprocal theorems, and one of them is called the inverse theorem of the other.

Every proposition has an inverse proposition, but not all theorems have inverse theorems.

Ruler compass drawing method

The method of drawing with ruler (without scale) and compass is called ruler drawing.

Basic drawing

The most basic and common ruler drawing is called basic drawing, which mainly includes the following:

(1) Make an angle equal to the known angle;

(2) bisecting the known angle;

(3) Make a point perpendicular to the known straight line;

(4) The median vertical line of the known line segment;

(5) A parallel line that intersects a point outside a straight line is called a known straight line.

related notion

A triangle with two equal sides is called an isosceles triangle.

A triangle with three equilateral sides is called an equilateral triangle, also known as a regular triangle.

An isosceles triangle with a right angle is called an isosceles right triangle.

Both equilateral triangle and isosceles right triangle are special cases of isosceles triangle.

Related concepts of isosceles triangle

In an isosceles triangle, two equal sides are called waist, the other side is called bottom, the included angle between the two waists is called top angle, and the two angles on the bottom are called bottom angle.

Main properties of isosceles triangle

The two base angles are equal.

As shown in the figure, where Δ AB = AC, AB = AC, take the midpoint D of BC, followed by AD.

Easy to prove: Δ δAbd?δACD, ∴∠ B = ∠ C.

As shown in the figure, δδABC is an equilateral triangle,

Then, from AB = AC, we get ∠ B = ∠B=∠C,

From ca = CB, ∠ A = ∠A=∠B,

So ∠ A = ∠ B = ∠ C, but ∠ A+∠ B+∠ C = 180,

∴∠A=∠B=∠C=60

As shown in the figure, AB = AC and ∠BAC in ABC are equally divided by AD.

Then through Δ δABD?δACD,

Available BD = CD, ∠ ADB = ∠ ADC,

But ∠ ADB+∠ ADC = 180,

∴∠ ADB = 90, so AD⊥BC,

Two other important inferences can be drawn from this.

Two important inferences

The bisector of the vertex of isosceles triangle is vertical and bisects the bottom;

The internal angles of equilateral triangles are equal, all equal to 60.

Another discussion method of isosceles triangle properties and its inference

In a triangle, equal sides have equal angles.

The bisector of the top angle, the midline and the height of the isosceles triangle are combined into one.

The decision theorem of isosceles triangle and the core of its two inferences can be summarized as equilateral. Are important methods to prove that two line segments are equal.

Inference 3

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

It is easy to prove that the inverse proposition of this inference is also correct. That is to say, in a right triangle, if a right-angled side is equal to half of the hypotenuse, then the angle opposite to this right-angled side is equal to 30.

use

It is easy to prove the conclusion by using the judgment theorem and property theorem of isosceles triangle: "in a triangle, if two sides are not equal, the angles of their pairs are not equal, and the angles of the big side pairs are also larger;" On the other hand, in a triangle, if two angles are not equal, then the sides they face are not equal, and the side opposite the big angle is larger. "

Symmetry axis and center

The perpendicular bisector of a line segment divides the line segment into two equal parts.

The midpoint of a line segment is its center. In the future, we should learn that "a line segment is a central figure symmetrical about the midpoint".

A line segment is a figure whose axis of symmetry is the middle vertical line.

The Inverse Theorem of the Unity Theorem of Three Lines

As shown in the figure, the geometric language of the property theorem of the vertical line in the line segment is:

So it can be used to determine isosceles triangle, and its theorem is essentially

Inverse theorem of three-line unity theorem.

"Distance" is different, so is "heart".

The distance in the property theorem and inverse theorem of vertical line in line segment refers to the distance between two points, and the distance in the property theorem and inverse theorem of angular bisector refers to the distance from pointing to a straight line.

The bisectors of the three angles of a triangle intersect at a point, and the distance from the point to the three sides is equal (this point is called the heart of the triangle).

The perpendicular lines of the three sides of a triangle intersect at a point, and the distance from the point to the three vertices is equal (this point is called the outer center of the triangle).

Important trajectory

As shown in figure (a). Distance to both sides of corners OA and OB

P3 P2 equidistant points P 1 ... form rays.

Line OP is a collection of points.

As shown in figure (b), the distance to the two endpoints of the line segment AB

All equal points P 1, P2, P3… form a straight line.

Line P 1P2, so this straight line can be regarded as a moving point shape.

Into a "trajectory."

Section 13 Axisymmetric and Axisymmetric Graphics

Axial symmetry

Folding a graph along a straight line, if it can overlap with another graph, then the two graphs are said to be symmetrical about this straight line, also known as axial symmetry.

According to the definition, if the sum of two graphs is symmetric about the straight line L, then:

(1) and the size and shape of these two graphs are exactly the same.

(2) After one of the graphs is folded along L, the sum should completely coincide, and naturally the corresponding points in the two graphs should also coincide.

In fact, the straight line L is the median vertical line connecting the corresponding points in two axisymmetric figures. It is easy to obtain the following characteristics:

Two graphs that are symmetrical about a straight line are conformal.

Property 2 If two figures are symmetrical about a straight line, then the symmetry axis is the middle vertical line connecting the corresponding points.

Attribute 3 Two graphs are symmetrical about a straight line. If their corresponding line segments or extension lines intersect, then the intersection must be on the axis of symmetry.

It is not difficult to see that if the connecting line of the corresponding points of two graphs is vertically bisected by the same straight line, then the two graphs are symmetrical about this straight line.

axial symmetric figure

If a figure is folded along a straight line, and the parts on both sides of the straight line can overlap each other, then the figure is called an axisymmetric figure.

Differences and relations between axisymmetric graphics and axisymmetric graphics

differentiate

① Axisymmetric means that two figures are symmetrical about a straight line, while axisymmetric figures are symmetrical about a straight line.

(2) The corresponding points of axial symmetry are on two graphs respectively, and the corresponding points in the axial symmetry graph are all on this graph.

(3) the axis of symmetry in the axial symmetry may be outside the two figures, and the axis of symmetry in the axial symmetry figure must pass through this figure.

get in touch with

① After folding along a straight line, the two sides can completely overlap.

(2) If two axisymmetric figures are regarded as a whole, then the figure reflected by this whole is one.

Axisymmetric graphics; On the other hand, if two sides of an axisymmetric figure about the axis of symmetry are regarded as two.

Graphics, then the two graphics corresponding to these two parts are symmetrical about this symmetry axis.

Section 14 Pythagorean Theorem

right triangle

In a right triangle, the two acute angles are complementary. The two sides of the right angle are called right angles, and the opposite side of the right angle is called hypotenuse, which is the longest.

Isosceles right triangle

Isosceles right triangle is a special case of right triangle. It is also a special case of isosceles triangle. The two base angles of an isosceles right-angled triangle are equal to 45, the top angle is equal to 90, and the two equal right-angled sides are the waist.

pythagorean theorem

In a right triangle, the sum of squares of two right-angled sides A and B is equal to the square of hypotenuse C, which is the Pythagorean theorem.

Judging right triangle

If the lengths of the three sides of Δ ABC are A, B and C, respectively, and are satisfied, Δ ABC is a right triangle, where ∠ c = 90.

The Inverse Theorem of Pythagorean Theorem in Section 15

Inverse theorem of Pythagorean theorem

Pythagorean theorem is the property theorem of right triangle, and the inverse theorem of Pythagorean theorem is the judgment theorem of right triangle. That is, in △ABC, if A2+B2 = C2, △ABC is Rt△.

How to judge whether a triangle is a right triangle?

Find the biggest side first (such as C).

Verify whether c2 and A2+B2 are equal.

If C2 = A2+B2, then △ABC is a right triangle with ∠ C = 90. If C2 ≠ A2+B2, then △ABC is not a right triangle.

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* * * * Key technologies * * * *

Method 1: Prove "narrative in words"

Geometric proposition "method.

This kind of problem is more difficult to prove than the general geometric problem, but there are still some ideas and methods. Generally, the problem is analyzed as a whole first, and the analysis content is roughly divided into the following four points, and then it is gradually solved.

(1) Setting and conclusion of analysis proposition;

(2) Draw a picture by combining the question with the conclusion;

(3) Write out the known and verified problems by combining conclusions and figures;

(4) analyze the problem.

Method 2: Angle evaluation method of isosceles triangle

When solving the angle evaluation problem of isosceles triangle, all possible situations should be considered, and the situation that a triangle cannot be formed should be excluded. Especially when solving the sum, difference, multiplication and semi-relationship of line segments or angles, the synthesis method or decomposition method is often used, with the help of adding auxiliary lines.

Method 3: Determine whether the triangle

Right triangle method

To judge a right-angled triangle, we can use the inverse theorem of Pythagorean theorem, the nature of the midline of a line segment or the definition of a right-angled triangle. These methods need to be mastered and applied flexibly.

Method 4: Select questions.

Every step of the geometric drawing problem should be justified, so we are required to master the axioms and theorems we have learned. To master ruler and ruler drawing, you need to draw more and practice more.

Knowledge points: congruent triangles's judgment and nature

Methods: Analysis method.

Ability: the ability to analyze and solve problems

Difficulty: medium

Knowledge points: congruent triangles; internal bisector

Methods: Comprehensive method; decomposition method

Ability: the ability to analyze and solve problems;

Logical reasoning ability

Difficulty: medium difficulty.

Knowledge points: the nature of isosceles right triangle;

The midline property of the line segment; pythagorean theorem

Methods: Comprehensive method.

Ability: the ability to analyze and solve problems

Difficulty: medium difficulty.

Knowledge points: the nature of line segments

Flat method: number-shape combination method

Ability: spatial imagination;

Ability to analyze and solve problems

Difficulty: medium difficulty.

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Topic 1: Ask more questions for one question, draw more pictures for one question, and solve more questions for one question.

There are many ways to improve the ability to analyze and solve problems, one of which is to carefully design examples and exercises in textbooks and tap their potential. The examples and exercises in the textbook provide a rich source for the proposition of the senior high school entrance examination. They are rich in connotation, exemplary and enlightening in transforming knowledge into ability, and typical and representative in solving problems. If you are not satisfied with the answer, but go deep into it for further excavation and multi-directional exploration after solving it, you can not only get a series of new propositions, but also get rid of the "sea of questions" and achieve twice the result with half the effort. Moreover, we think about problems from different angles and directions and explore different solutions, thus broadening our thinking and cultivating the flexibility and adaptability of our thinking.

Topic 2: Improve the problem-solving ability by expanding, dissecting, stringing and modifying.

When learning geometry, I feel that the examples are easy to learn and understand, but I can't do anything about slightly changing and broadening the questions. The reason is that the learning of examples is regarded as an isolated problem, and the problem will be solved after learning, which leads to the lack of adaptability in solving problems. But if we can pay attention to the expansion, anatomy, series connection and adaptation of the problem in peacetime, we can solve this problem better.

1. extension: the original conditions are extended to make the conclusion more abundant and sufficient.

2. Anatomy: Analyze the original problem, dismember the complex figure into several basic figures, and make the problem hidden and obvious.

3. Tandem: Associating similar, similar and opposite problems from the form of examples (conditions, conclusions, etc.). ).

4. Adaptation: change the conditional form of the original question and explore whether the conclusion is valid.

Topic 3: Analysis, Synthesis and Auxiliary Lines

When we study the related problems of inequality, we will find many ingenious methods, constantly learning and mastering the mathematical thought of analogy, the thought of combining shape and number, and transforming the thought from unknown to known. By learning these changing problems, we can master the solutions of inequalities and inequality groups in an all-round way, so as to improve our ability to analyze and solve problems.

Topic 4: Some applications of inequality

In plane geometry, the main idea of proving the problem is: (1) analysis, that is, starting from the conclusion, pushing it back step by step until the known facts are obtained. (2) synthesis method, starting from known conditions, using formulas, theorems, properties, etc. What has been learned leads to the conclusion of proof. (3) organically combine the comprehensive method with the analytical method: on the one hand, we can infer from what is known and what conclusions can be drawn from what is known; On the other hand, "from the unknown to the need to know", from the conclusion to see what conditions are needed, once the knowledge and the need to know communicate, the idea of proof will be there. Adding auxiliary lines is an important means to prove geometric problems, and it is also one of the difficulties in learning.

Topic 5: There are two basic methods to prove geometric problems:

One is to proceed from conditions and gradually deduce through a series of established propositions until the problem of proof is reached. In short, this is a method derived from causality, which we call direct proof or synthesis. The steps of synthesis are as follows: prove AB, because AC, CD, …, X, and xB, AB.

The other is to assume that the conclusion of the proposition holds, consider what conditions are needed to achieve the goal, and go through a series of backward derivation until the known conditions are reached. In short, this is the way to catch the cause and effect, which we call analytical method. The procedure of proving the problem by analytical method is as follows: if you want to prove "AB", that is, BA, if you can analyze BC, CD, …, X, xA, you can assert BA, that is, AB.

In practice, these two methods are often used in combination. First of all, we analyze and explore the groundwork, and then solve the problem in an all-round way. In short, it means "push back and move forward".

—Translation, rotation and symmetry

In geometric proof, it is often necessary to transform a graph properly. Common geometric transformations include congruence transformation, equal product transformation and similarity transformation.

This chapter only talks about congruent transformation, that is, the transformation that only changes the position of the figure without changing the shape and size of the figure.

There are three common forms of congruence transformation:

1. Translation: Move some line segments or even the whole figure to an appropriate position in parallel, make an auxiliary figure, and make the problem clear.

To solve it. The basic feature of translation is that any line segment has been translated.

In this process, its length remains the same.

2. Rotation: the plane figure is rotated by a fixed angle α around a certain point m on the plane to obtain a figure with the same shape and size as the original figure, so that,

The transformation of is called rotation transformation, m is called rotation center, and α angle is called rotation.

Corner.

The main properties of rotation transformation are: (1) the transformed graph is the same as the original graph; (2) The angle formed by any line segment in the original image and the rotated corresponding line segment is equal to the rotation angle.

3. Symmetry: Rotate a graph (or part of it) around a straight line by 180 to get a graph with the same shape and size as the original graph. This transformation is called axisymmetric transformation. The main feature of axisymmetric transformation is that the axis of symmetry is the median vertical line connecting all corresponding points before and after turning.

In addition to axial symmetry, there is also central symmetry, which we will talk about in the next chapter quadrilateral.

Method summary:

Complex graphics are composed of simple basic graphics, so complex graphics can be decomposed into several basic graphics, so the problem is obvious.

When it is difficult to prove the problem directly, the basic graphics are often constructed by adding auxiliary lines to solve the problem.

Comprehensive method is to explore ways to solve problems from known conditions.

The method of analysis is to find a way to prove the idea by reverse deduction from the conclusion.

The method of "combining the two ends", that is, comprehensively using the above two methods, can find the proof idea. (also called analytical synthesis).

Transforming thinking is to transform complex problems into simple ones; Or the idea of turning an unfamiliar problem into a familiar one.