What are the commonly used auxiliary lines in middle school geometry?

1. See the median line at the midpoint, and double the length of the midline.

In geometry problems, if you give a midpoint or a midline, you can consider using the midpoint as a median line or doubling the midline to solve the problem.

2. In the proof of proportional line segments, parallel lines are often used.

Parallel lines are often used to retain one ratio in the conclusion and then link it to another ratio in the conclusion through an intermediate ratio.

3. For the trapezoidal problem, the commonly used methods for adding auxiliary lines are: 1. The two ends of the upper bottom are perpendicular to the lower bottom.

2. Make a waist parallel line through one end of the upper bottom.

3. Make a diagonal parallel line through one of the ends of the upper bottom.

4. The midpoint of one waist is used as a parallel line of the other waist.

5, through the upper bottom of the end of the end of the waist and a waist of the straight line intersects with the extension line of the lower bottom 6, the trapezoidal median line.

7 Lengthen the loins so that they meet.

Fourth, in solving the problem of the circle.

1. Two circles intersect and connect common chords.

2 The two circles are tangent, and the tangent is introduced through the tangent point.

3. See the diameter and think at a right angle.

4. In case of tangent problems, the radius connecting the tangent points is a common auxiliary line.

5. When solving problems related to strings, the center distance of the strings is often made.

The above is my summary of the common auxiliary lines.

In fact, the auxiliary lines commonly used in geometry are not certain. It generally depends on the topic.

Many of the auxiliary lines of the topic are essential in the train of thought you prove.

When you think about that, you naturally add auxiliary wires.

Seventh grade used, but in smaller amounts, from isosceles triangles.

At the beginning, the auxiliary lines gradually increased.

Auxiliary line refers to a straight line or line segment with great value made on the basis of the original drawing, which is mostly usedGeometryto troubleshoot difficult geometry issues.

When elementary school students are looking for the ratio of area or area, in fact, some of the questions use auxiliary lines.

Junior high school geometry will generally be used after learning the properties of parallel lines and judging theorems (generally the second volume of the seventh grade); In the first book of the eighth grade, students will learn congruent triangles.

After that, it will be applied more.

Expansion: 1Transform scattered geometric elements into relatively concentrated geometric elements (e.g., concentrating scattered elements in a triangle or two congruent triangles so that the theorem can be applied to the application).

2.Transform irregular graphics into regular graphics, and transform complex graphics into basic graphics of Jianchang family single.

3.In planar geometry, auxiliary lines are represented by dashed lines. Solid geometry.

, the visible is represented by a solid line, and the invisible is represented by a dotted line.

In junior high geometry, you can have angular bisector lines in the diagram and make perpendicular lines to both sides as auxiliary lines.

Triangle with angular bisector in the figure, which can be perpendicular to both sides. You can also fold the graph in half, and the relationship between symmetry and symmetry will appear. Angles bisector parallel lines, isosceles triangles to add.

Angular bisector line plus perpendicular line, three lines in one to try. The line segment bisects the line vertically, often connecting the lines to both ends. The line segment and the difference are half times, and the extension and shortening can be tested.

The line segment and the difference inequality are moved to the same triangle. There are two midpoints in the triangle, and when they are connected, they form a median line. There is a midline in a triangle, and the midline of the double length is congruent.

Quadrilaterals, parallelograms appear, symmetrical centers equisect points. The trapezoidal problem is cleverly transformed into a triangle or a flat four. Touching the years to move the waist horizontally, move the diagonal, and the two waists are extended to make a height.

If there is a midpoint of the waist, carefully connect the median line. The above method does not work, and the midpoint of the waist is made in equal proportions. The certificate is similar, than the line segment, and it is customary to add parallel lines.

Equal to the proportion of the Lingwu formula, it is very important to smile and close your eyes to find the line segment. It is directly proved that there is difficulty, and the same amount of substitution is less troublesome. A high line is made above the hypotenuse, and a large piece of the middle item is proportional.

How to draw basic type guides:

For questions about the middle line of a triangle, the middle line is often doubled. In the case of a midpoint, the median line of a triangle is often used, and the problem can be easily solved by appropriately shifting the conclusion of the evidence. For problems containing bisector lines, the angle bisector is often used as the axis of symmetry, and the properties of the angle bisector and the condition in the problem are used to construct congruent triangles, so as to use the knowledge of congruent triangles to solve problems.

The conclusion is that the problem of two equal line segments often draws auxiliary lines to form congruent triangles, or uses some theorems about bisector line segments. The conclusion is that the sum of a line segment and another line segment is equal to the third line segment, and the truncation method or the shortening method is often used, and the so-called truncation method is to divide the third line segment into two parts, proving that one part of it is equal to the first line segment, and the other part is equal to the second line segment.

Everyone says that geometry is difficult, and the difficulty lies in the auxiliary line. How do I add an auxiliary line? Grasp theorems and concepts.

It is also necessary to study assiduously and find out the rules based on experience. There are angular bisectors in the diagram, which can be perpendicular to both sides.

Angles bisector parallel lines, isosceles triangles to add. Angular bisector line plus perpendicular line, three lines in one to try.

The line segment bisects the line vertically, often connecting the lines to both ends. There are two midpoints in the triangle, and when they are connected, they form a median line.

There is a midline in the triangle, and the prolongation comma refers to the contour of the long midline. A parallelogram appears, symmetrically centrically bisecting points.

Make a high line inside the trapezoid, and try to pan it around the waist. It is common to move diagonal lines in parallel and make up triangles.

The certificate is similar, than the line segment, and it is customary to add parallel lines. For equal area sub-proportional exchange, it is very important to find the line segment by looking for the finger core.

It is directly proved that there is difficulty, and the same amount of substitution is less troublesome. A high line is made above the hypotenuse, and the proportion is a chain with a large piece.

The radius is calculated with the chord length, and the chord centroid distance comes to the intermediate station. If there are all lines on the circle, the tangent points are connected with the radius of the center of the circle.

The Pythagorean theorem is the most convenient for the calculation of the tangent length. To prove that it is a tangent, the radius perpendicular line is carefully identified.

It is a diameter and forms a semicircle and wants to form a right-angle diameter chord. The arc has a midpoint and a central circle, and the vertical diameter theorem should be memorized.

The two chords on the periphery of the corner, the diameter and the end of the chord are connected. The string is cut to the edge of the tangent string, and the same arc is diagonally to the end.

If you encounter intersecting circles, don't forget to make common chords. Two circles tangent inside and outside, passing through the tangent point of the tangent line.

If you add a connecting line, the tangent point must be on it. The auxiliary line is a dotted line, and you should be careful not to change it when drawing.

Basic drawing is very important, and you must be proficient in mastering it at all times. It is necessary to be more attentive to solving problems, and often summarize the methods.

Don't blindly add lines, and the method should be flexible and changeable. Analyze and choose comprehensive methods, no matter how many difficulties there are, they will be reduced.

With an open mind and hard work, the grades rose into a straight line.

Summary of the practice of geometric guides.

How to do common guides in triangles:

Extending the midline to construct congruent triangles;

Use folds to construct congruent triangles;

Parallel lines construct congruent triangles;

Make a line to construct an isosceles triangle.

There are several common ways to do the guides:

1) When encountering an isosceles triangle, you can make the height on the bottom edge, and use the property of "three lines in one" to solve the problem, and the thinking mode is the "fold in half" in the congruent transformation.

2) When encountering the midline of the triangle, doubling the length of the midline, making the extension segment equal to the length of the original midline, constructing a congruent triangle, using the thinking mode is the "rotation" in congruent transformation.

3) When encountering an angle bisector, you can make perpendicular lines from a certain point on the angle bisector to both sides of the angle, and the thinking mode used is the "folding" in the congruence transformation of triangles, and the knowledge points tested are often the property theorem or inverse theorem of the angle bisector.

Figure 1. 4) Construct a congruent triangle by making a specific bisector at a certain point on the graph, using the thinking mode of "translation" or "flipping and folding" in congruent transformations.

5) The truncation method and the shortness method, the specific method is to intercept a line segment on a certain line segment and be equal to a specific line segment, or extend a certain line segment, which is equal to a specific line segment, and then use the relevant properties of triangle congruence to explain This method is suitable for proving the problem of sum, difference, multiple, and classification of line segments.

Special method: When solving problems such as the fixed value of triangles, the line segments from a certain point to the vertices of the original triangle are often connected, and the knowledge of the triangle area is used to solve them.

Figure II.

The techniques for solving the problems of the secondary two mathematics and geometry auxiliary lines are as follows:

In the triangle diagram, there are angular bisector lines, and the code excitation can be perpendicular to both sides. It can also be Chi Xiang socks will be folded in half to see, symmetrical relationship after the present. Angles bisector parallel lines, isosceles triangles to add. Angular bisector line plus perpendicular line, three lines in one to try.

Intercept constructive congruent AB cd, be bisected abc, ce bisect bcd, point e is on ad, verify: bc=ab+cd.

Analysis: In this problem, BF=AB can be intercepted on the long line segment BC, and then CF=CD can be proved to achieve the purpose of proof. The angular bisector is used to construct congruent triangles.

The points on the angular equinox line are perpendicular to both sides, congruent is known> ab ad, bac= fac, cd=bc. Certificate: ADC + b = 180

Proving the type of questions about line segments and inequalities is also very common in exams, usually using the trilateral relationship theorem of triangles.

In a triangular feast, the sum of the two sides is greater than the third side, and the difference between the two sides is less than the third side.

Then the line segment that is not in a triangle is added by the method of truncation and complementation, and the transformation of the line segment is obtained through the congruence proof of the triangle. In this way, the relationship between the sum and difference of the line segment can appear in a triangle.