When proving and solving geometric problems, if it is an axisymmetric figure, in order to make full use of the properties of axisymmetric figures, it is often necessary to add an axis of symmetry. For example, isosceles triangles often add the bisector of the top angle; The problem of rectangle and isosceles trapezoid is often to add a connecting line between the opposite midpoint and the connecting line between the two bottom midpoints; Slant lines are often used in square and diamond problems. In addition, if the graphics encountered are not axisymmetric, a straight line is often selected as the symmetry axis and added as the symmetry axis, or the graphics on one side of the axis are reflected to the other side by folding, thus realizing the relative concentration of conditions. Using the test case 1, it is known that there is a certain point outside the straight line, so try to find two points on it to make it (fixed length). It is also the shortest. Analysis: When the point is translated in the direction (as shown in figure 1), then the problem is transformed into finding a point on the ground to make it shortest. Practice: over-acting, doing, doing about the symmetry point, and connecting to B. Working on the ground, doing one thing, doing two things. Proof: take another point on the ground, bang, bang'. Then PA'=P'A', P'B'=P'B', PA'B'C' is a parallelogram, ∴ CB' = pa'. ∫c ' b+b ' p > CP '，∴pa '+p ' b ' & gt； Pa+Pb。 Example 2 is shown in Figure 2. In △ABC, it is a point on the bisector of △ A. Verification: Pb+PC >; AB+AC。 Analysis: Because the bisector of the angle is the symmetry axis of the angle, make the symmetrical graph AD of AC about AP and connect DP and CP, then DP=CP, BD = AB+AC. In this way, AB+AC, PB and PC are concentrated in △BDP, so Pb+PD >: BD can be used as PB+PC >; AB+AC。 Proof: (omitted). Comments: After turning into an axisymmetric figure, it plays the role of a relatively concentrated condition, and straightens the broken line (for example, AB+AC pulls it to BD). Example 3 Diagonal lines of isosceles trapezoid are perpendicular to each other, and the lines are equal. Find the height of this trapezoid. Solution: Let an isosceles trapezoid AD∨BC, AB =. The middle line ef = m. The middle points m and n passing through AD and BC are straight lines, and the isosceles trapezoid ABCD forms an axisymmetric figure with respect to the straight line MN. Point ∴O is on MN, OA=OD, OB=OC, AM=DM, BN = CN. It is also AC⊥BD, so △AOD and △BOC are isosceles right triangles. ∴20m+2on = 2m。 ∴ OM+ON =, so the trapezoidal height Mn = m. The application example of determining the direction is 1, as shown in figure 1. The quadrilateral ABCD is a rectangular billiard table with black and white balls at E and F respectively. How to play the black ball e so that the black ball hits the DC on the table first? Solution: Let point E be the symmetrical point E' about the straight line CD, which is connected with FE'. The intersection point p with CD is the impact point, and the point p is the demand. Example 2 is shown in fig. 2. Car A travels right along Highway L from place A, and car B starts from place B. The speed of car B is the same as that of car A. Car B should catch up with car A in the shortest time. What is the direction of B car? Solution: Let the median vertical line EF of AB intersect with the straight line L at point C, and the car B can travel in the direction of BC. Determine the position of the point and find the minimum value. Example 3 as shown in figure 3, ab∑CD, AC⊥CD, find a little E on AC to minimize BE+DE. Solution: Let point B be the symmetrical point B ′ about AC, connecting DB ′, and intersecting with AC at point E. I want to build a branch office D on L 1 highway and a branch office E on L2 highway to minimize the sum of AD+DE+EA. Solution: Regarding the symmetrical points B and C of L 1 and L2, make point A, connect BC, cross L 1 at point D, and cross L2 at point E, and points D and E are the required points. Example 5 Example 5

Analysis: Let the pump station be repaired at point C. The essence of this problem is to find the shortest length of polyline AC+BC. We can make the symmetrical point A' of point A about straight line L, as shown in figure 1. According to the symmetry, AC+BC=A'C+BC, so the straight line L connecting BA' is at point C, and point C is the position of the pump station, because the length AC+CB of the broken line is converted into a line segment. According to the shortest line segment between two points, we can determine that point C is the position of the water pump. Combined with other disciplines, ten Buddhist temples were built in a certain place in the Tang Dynasty. When it was completed, Fu Yin wrote a couplet on the right side of the temple gate, "Ten thousand bricks and hundreds of craftsmen make ten Buddhist temples", hoping that someone would make a couplet and express it appropriately. Can you make couplets? Couplets are ten thousand, thousand, hundred and ten. After a few months, no one can tell the truth. Scholar Li Sheng passed by and felt that there were no couplets in front of the temple. He is very emotional. He thought hard in front of the temple for several days, but failed to make couplets. Once he went to the front of the temple and saw a big ship coming from a distance, and the boatman was rowing hard. At this moment, Li Sheng suddenly had a brainwave and went out. Repeatedly praised "Vivi". This couplet is so harmonious in numbers, beautiful in things to things and symmetry. It can be seen that the beauty of symmetry is also vividly and profoundly reflected in literature. Axisymmetry is everywhere in life. As long as you are good at observing it, you will find the rich cultural value contained in it and the endless enjoyment brought by the beauty of symmetry. Solving the minimum problem by solving the equation after symmetry often changes the thinking angle by using the position of symmetrical thinking, and then

Example 1 Two points A (0 0,2) and B (4, 1) are known. Point P is a point on the X axis, and the value of PA+PB is the smallest. Analysis: As shown in figure 1, first mark the positions of point A and point B in the coordinate system, and then determine point P on the X axis, so that PA+PB should be considered). There are three villages A, B and C on the same side of a highway. It is necessary to build a freight station D along the highway to transport agricultural materials to villages A, B and C. The route is D→A→B→C→D or D → C → B → A → D Draw three points A, B and C, and get the coordinates of point D. Analysis: Assuming that point D has been determined, the sum of transportation distances is DA+AB+BC+CD.