The practice of common auxiliary lines in junior high school geometry, and the practice of auxiliary

People say that geometry is difficult, and the difficulty lies in the auxiliary lines.

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.

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.

It is necessary to prove that the line segment is doubled and halved, and the extension and shortening can be tested.

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 extension of the midline is an isomidline.

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 line segments.

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.

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.

To prove that it is a tangent, the radius perpendicular line is carefully identified.

The arc has a midpoint and a central circle, and the vertical diameter theorem should be memorized.

To make a circumscribed circle, make a perpendicular line on each side.

It is also necessary to make an inscribed circle, and the bisector of the inner angle dreams come true.

If you encounter intersecting circles, don't forget to make common chords.

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.

How can this be explained, you have to focus on the specific topic to know. However, I think the most commonly used auxiliary lines: extensions, diagonals, midlines, perpendiculars, and angular bisectors.

Generally speaking, the more difficult problems are used in combination with more than two auxiliary lines. Pay attention to the summary when you usually do the questions. would be useful.

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.

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, often make string center distances.

The above is my summary of the common auxiliary lines.

The formula for the middle school geometry auxiliary line is as follows:

1. Triangle.

There are angular bisectors in the diagram, 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 change the difference inequality, move to the same triangle.

2. Quadrilateral.

A parallelogram appears, symmetrically centrically bisecting points. The trapezoidal problem is cleverly converted to and . Translate the waist, move the diagonal, and extend the waist to make it high.

If there is a midpoint in the waist, carefully connect to 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, adding the line parallel into the habit of no loss.

For equal area sub-proportional exchange, it is very important to find line segments.

3. Round. 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 length of the tangent is calculated by the Pythagorean theorem which is the easiest to check. 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. Two chords on the periphery of the corners, with diameters and chord endpoints connected.

The string is cut to the edge of the tangent string, and the same arc is diagonally and so on. To make a circumscribed circle, make a perpendicular line on each side. It is also necessary to make an inscribed circle, and the bisector of the inner angle dreams come true.

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.

It is necessary to add a circle at equal angles to prove that the topic is less difficult.

Principle of adding lines:

1. Transform scattered geometric elements into relatively concentrated geometric elements (e.g., concentrate scattered elements in a triangular no-loss shape or two congruent triangles, so that the theorem can be applied).

2. Transform irregular graphics into regular graphics, and transform complex graphics into simple basic graphics.

3. In plane geometry, auxiliary lines are represented by dashed lines. In solid geometry, what is visible is represented by solid lines, and what is invisible is represented by dashed lines.