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.