1. In the triangle ABC, the angle ABC is 60 degrees, and AD and CE divide the angles BAC and ACB equally. What is the quantitative relationship among AC, AE and CD? 2. Divide each side of an equilateral triangle into three parts and grow a small equilateral triangle with one third of the original side, which is called primary growth. If it is tripled, the area of the polygon is several times that of the original triangle. (This line is called Euler line) It is proved that the midpoint of three sides of the same triangle, the vertical foot of three perpendicular lines, and the midpoint of the line segment from each vertex to the vertical center is a 9-point * * * circle. ~ ~ (this circle is called the nine o'clock circle) 3. It is proved that for any triangle, there must be two sides A and B, the ratio of A to B is greater than or equal to 1, and the root less than 2 is 5 plus14. It is known that the three heights of △ABC intersect at the vertical center O, where AB=a, AC = B and ∠ BAC = α. Please use a formula containing only three letters A, B and α to express the length of AO (not all three letters can be used up, but other letters must not be used). 5. Let the straight line be y=kx+b (k, b is constant. K is not equal to 0). It must pass the intersection of x-y+2=0 and x+2y- 1=0 (-1, 1). So b = k+65438. 2) The straight line perpendicular to x-y+2=0 is y=-x+2 (2). The intersection of line (2) and line (1) is a, and the intersection of line (2) and line x+2y- 1=0 is b, so the midpoint of AB is (0. Let angle APB= angle BPC= angle CPA, and PA=8 PC =6, then PB= 2 P is a point in the rectangular ABCD, and PA=3 PB= 4 PC=5, then PD= 3, triangle ABC is an isosceles right triangle, angle C = 90 O is a point in the triangle, and the distance from point O to each side of the triangle is equal to 1. Rotate the backup point o of triangle ABC clockwise by 45 degrees to get the common part of triangle A 1B 1C 1. It is proved that triangle AKL, triangle BMN and triangle CPQ are isosceles right triangles. 2) find triangle ABC and triangle A 1b 1c65448. It is known that triangles ABC, A, B and C have three sides respectively. It is proved that the sum of squares of three sides of a triangle is greater than or equal to the root number 3 of16th degree (that is, the root number 3 of a2+b2+c2 is greater than or equal to16th degree). Exercise 1. Multiple choice questions 1. If α and β are internal angles on the same side and α = 50. Then β is equal to () (a) 55 (b) 125 (c) 55 or 125 (d), which cannot be determined. 2. as shown in figure 19-2-(2) AB‖CD if ∠2 is ∞. ∠2 is equal to () (a) 60 (b) 90 (c)120 (d)1503. As shown in figure 19-2-(3)∠ 1+∞. Then ∠4 degrees () (a) equals ∠1(b)110 (c) 70 (d) cannot be determined. 4. as shown in figure 19-2-(3) ∠ 65436. Then the number of ∠ 1 is () (a) 70 (b)10 (c)180-∠ 2 (d), which is incorrect. 5. as shown in figure 19-2 (. Then () (a) ≈1= ∠ 2 (b) ≈ 2 = ∠ 3 (c) ≈1= ∠ 4 (d) ab ∠ CD 6. As shown in figure 65434. ∠ whether the bed is () (a) acute angle (b) right angle (c) obtuse angle (d) cannot be determined. 7. If one side of two angles is on the same straight line and the other side is parallel to each other, then the relationship between the two angles is () (a) equal (b) complementary (c) equal and complementary (d) equal or complementary. 8.∠ α = () (a) 50 (b) 80 (c) 85 Answer: 1 .d2.c3.c4.c5.d6.b7.d8.bThe final question of the second semester of the first grade geometry1. One is obtuse B. Both are obtuse C. Both are right angles D. There must be a right angle 2. If ∠ 1 and ∠2 are adjacent complementary angles, and ∠ 1 > ∠2, then ∠2' s complementary angle is () 3. The following statement is correct. Then these two angles must be adjacent complementary angles. D two known points outside and on the straight line are perpendicular to the known straight line. 4. In the same plane, the positional relationship between two non-overlapping straight lines may be () A. Parallel or intersecting B. Vertical or parallel C. Vertical or intersecting D. Parallel, vertical or intersecting 5. If two non-adjacent right angles have a common edge, then the other edge is mutually () A. Parallel B. Vertical C. On the same straight line D. Parallel or vertical.