If it is proved that two straight lines can extend vertically, the intersection angle is 90; It is proved that the doubling relation of line segment can take the midpoint of line segment or double half line segment; It can also be similar to adding auxiliary lines to prove the relationship between times and half of angles.

2 according to the basic graphics to add auxiliary lines:

Every geometric theorem has its corresponding geometric figure, which we call basic figure. Adding auxiliary lines often has the nature of basic graphics, which can be supplemented when the basic graphics are incomplete, so "adding lines" should be called "supplementing graphics"! This can prevent adding lines indiscriminately, and adding auxiliary lines has rules to follow. Examples are as follows:

(1) parallel lines are a basic figure:

When parallel lines appear in geometry, the key to adding auxiliary lines is to add a third line intersecting with the two parallel lines.

(2) The isosceles triangle is a simple basic figure:

In geometric problems, when there are two equal line segments from one point, it is often necessary to complete the isosceles triangle. When the combination of bisector and parallel line appears, the intersection of two sides of parallel line and angle can extend to form an isosceles triangle.

(3) The important line segment in the isosceles triangle is an important basic figure:

The midpoint on the bottom of the isosceles triangle is added to the midline on the bottom; When the bisector of the angle is combined with the vertical line, when the vertical line intersects with the two sides of the angle, the basic figure of the important line segment in the isosceles triangle can be extended.

(4) The basic figure of the center line on the hypotenuse of the right triangle.

The midpoint on the hypotenuse of a right triangle is often added to the midline on the hypotenuse. If the line segment is the hypotenuse of a right triangle, it is necessary to add the midline on the hypotenuse of the right triangle to get the basic figure of the midline on the hypotenuse of the right triangle.

(5) The basic figure of triangle midline

When there are multiple midpoints in geometry problems, the basic figure of triangle midline is often added to prove it. When there is a midpoint without a midline, add a midline. When there is a midline triangle that is incomplete, you need to add a complete triangle. When there is a line segment folding relationship, and the line segment with the same end point has a midpoint, the parallel lines folded by the line segment can be added through the midpoint to get the basic figure of the triangle midline; When there is a line segment folding relationship, and the endpoint of the line segment is the midpoint of a line segment, the basic figure of the triangle midline can be obtained by adding the parallel lines of the line segment with the midpoint.

(6) congruent triangles:

Congruent triangles has axis symmetry, center symmetry, rotation and translation. If two equal line segments or two equal angles are symmetrical about a straight line, you can add an axisymmetrical congruent triangles: or add an axis of symmetry, or flip a triangle along the axis of symmetry. In geometric problems, when one or two groups of equal-length line segments are located on both sides of a pair of vertex angles and on a straight line, it can be proved by adding a centrosymmetric congruent triangles. The addition method is to connect four endpoints in pairs or increase parallelism through two endpoints.

(7) similar triangles:

Similar triangles has parallel lines (similar triangles of parallel lines), intersecting lines and rotational transformation; When the line segments overlap on a straight line (the midpoint can be regarded as the ratio of 1), the parallel line similar triangles can be added. If parallel lines pass through endpoints, the line segments that can be divided into points or other endpoints are parallel. There are often many shallow line methods for this kind of problem.

(8) Right triangle with special angle

When special angles of 30 degrees, 45 degrees, 60 degrees, 135 degrees and 150 degrees appear, a right triangle with special angles can be added, and the ratio of three sides of a 45-degree right triangle is1:1:√ 2; It is proved that the ratio of three sides of a right triangle with an angle of 30 degrees is 1:2:√3.

(9) Circumferential angle on a semicircle

The diameter and the point on the semicircle appear, plus the circumferential angle of 90 degrees; The appearance of 90-degree circumferential angle increases its relative chord diameter; In plane geometry, there are only more than twenty basic figures, just like a house is composed of anvil, tiles, cement, lime, wood and so on.

Junior high school mathematical geometry problem-solving skills II. Draw auxiliary lines of basic graphics

Auxiliary line addition method for 1. triangle problem

Methods 1: The midline of triangle is always double. Questions with a midpoint, usually the center line of a triangle. By this method, the conclusion to be proved is properly transferred, and the problem is easily solved. Method 2: When there is a bisector, we often take the angular bisector as the symmetry axis, use the properties of the angular bisector and the conditions in the problem to construct a congruent triangles, and use congruent triangles's knowledge to solve the problem.

Method 3: The conclusion is that when two line segments are equal, auxiliary lines are often drawn to form congruent triangles, or some theorems about bisecting line segments are used.

Method 4: The conclusion is that the sum of one line segment and another line segment is equal to the third line segment, and truncation method or complement method is often used. The so-called truncation method is to divide the third line segment into two parts and prove that one part is equal to the third line segment.

The first line segment and the other part of the second line segment are equal.

2. Addition of commonly used auxiliary lines in parallelogram

The two groups of opposite sides, diagonal lines and diagonal lines of parallelogram (including rectangle, square and diamond) all have some similar properties, so there are some similarities in the method of adding auxiliary lines. The purpose is to create parallelism and verticality of line segments, form congruence and similarity of triangles, and turn parallelogram problems into common problems such as triangles and squares. Common methods are as follows, for example:

(1) diagonal or translation diagonal:

(2) Take the vertex as the edge and construct a right triangle with vertical lines.

(3) Connect the diagonal intersection point with the midpoint of one side, or take the parallel line intersecting the diagonal intersection point as one side, and construct a line segment parallel line or midline.

(4) Connect the vertex with a point on the opposite side or extend this line to form a triangle with similar or equal products.

(5) The vertical line intersecting the vertex diagonally constitutes a parallel line segment or triangle congruence.

3. The method of adding commonly used auxiliary lines of trapezoid.

Trapezoid is a special quadrilateral. It is the synthesis of parallelogram and triangle knowledge, and can be solved by adding appropriate auxiliary lines to turn the trapezoid problem into a parallelogram problem or a triangle problem. The addition of auxiliary lines becomes a bridge to solve problems. Auxiliary lines commonly used in trapezoid are:

(1) translates a waist in the trapezoid.

(2) Translating a waist outside the trapezoid.

(3) Translating the two waists in the trapezoid.

(4) Stretch the waist.

(5) The bottom of the trapezoidal upper sole is downward.