First of all, there are two ways to add auxiliary lines:
1. Add auxiliary lines as defined:
If it is proved that the perpendicularity of two straight lines can be extended to make their intersection angle 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. Add auxiliary lines according to the basic graphics:
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, congruent triangles with symmetrical center can be added to prove it. Addition is to connect four endpoints in pairs or add parallel lines 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 are added at the endpoints, points or line segments at other endpoints can be divided into parallel directions. 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.
Second, the drawing method of basic graphic auxiliary lines
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 first line segment and the other part is equal to the second line segment.
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 transform 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) Pass through both ends of the trapezoid upper bottom to raise the bottom.
(6) Translation diagonal
(7) Connect a vertex of the trapezoid with the midpoint of a waist.
(8) The midpoint of one waist is the parallel line of the other waist.
(9) As the center line
Of course, in the proof and calculation of trapezoid, the added auxiliary line is not necessarily fixed and single. By bridging the auxiliary lines, the trapezoidal problem is transformed into a parallelogram problem or a triangle problem, which is the key to solve the problem.
4. Add common auxiliary lines in the circle
In plane geometry, when solving problems related to circles, it is often necessary to add appropriate auxiliary lines to connect the problems and conclusions, so that the problems can be solved naturally. Therefore, mastering the general rules and methods of auxiliary lines flexibly is of great help to improve students' ability to analyze and solve problems.
(1) regards the chord as the distance from the center of the chord.
For the chord problem, the chord center distance (sometimes the corresponding radius) is often made, and the connection between the topic and the conclusion is communicated through the vertical diameter bisection theorem.
(2) Take the diameter as the angle of a circle.
If the diameter of a circle is known in the topic, it is generally the circumferential angle opposite to the diameter, and the problem is proved by using the characteristic that the circumferential angle opposite to the diameter is a right angle.
(3) regard tangent as radius
The condition of the proposition includes the tangent of the circle, which is often the radius connecting the tangent points. This paper proves this problem by using the property that the tangent is perpendicular to the radius.
(4) The tangents of two circles are common tangents.
For the problem of tangency between two circles, the common tangent of two circles or their connecting lines is generally made by the tangent point, and the relationship between the angles related to the circles can be obtained by the common tangent.
(5) The intersection of two circles is a chord.
For the problem that two circles intersect, it is usually to make a chord. The chords of two circles can be connected by common chords, and the peripheral angle or central angle in two circles can be connected.