2. Add auxiliary lines according to the basic figure: every geometric theorem has its corresponding geometric figure, which we call the basic figure. Adding auxiliary lines often has the nature of basic graphics, and the basic graphics are supplemented when the basic graphics are incomplete, so "adding lines" should be called "mending drawings"! 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) An isosceles triangle is a simple basic figure: when there are two equal line segments from one point in a geometric problem, it is often necessary to complete the isosceles triangle. When the bisector of the angle is combined with the parallel line, the parallel line can extend to intersect with both sides of the angle 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 with 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 midline on the hypotenuse of the right triangle is often added to the midpoint of the hypotenuse when the basic figure of the hypotenuse of the right triangle appears. 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) When there are multiple midpoints in the geometric problem of the triangle centerline basic figure, the triangle centerline basic figure is often added to prove it. When there is a midpoint without a midline, increase the midline. When a triangle with a midline is incomplete, a complete triangle is needed. 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 axial symmetry, central 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 with parallel lines), intersecting lines and rotation types; When the comparison line segments overlap on a straight line (the midpoint can be regarded as the ratio of 1), the similar triangles of parallel lines 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) Special Angle Right Triangle When there are special angles of 30 degrees, 45 degrees, 60 degrees, 135 degrees and 150 degrees, a special angle right triangle can be added, and the ratio of three sides of the 45-degree right triangle is1:√ 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) The circumference angle on the semicircle is the same as the diameter of the point on the semicircle. If the circumference angle of 90 degrees is added, the relative chord diameter is added; 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.