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HW Exercise 1.3 (#16 Constructions)

HW Exercise 1.3 (#16 Constructions)
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2017-01-25 00:32
CircleFunctionCoordinate GeometryGeometric Construction
① This example demonstrates a comprehensive geometric construction involving segments, lines, circles, perpendicular lines, and their intersections, resulting in a complex figure built step by step. ② Core mathematical concepts include geometric construction (compass-and-straightedge ideas), quadratic functions, quadrilateral properties, coordinate geometry, and curve sketching; the original tags also mention LCM and GCD, suggesting a possible number-theoretic context. ③ Construction highlights: from the summary, GeoGebra commands such as Segment, Line, Intersect, Circle, Point, and OrthogonalLine are used, creating numerous point objects (A, B, C, D, E, F, G, H, I, J, K, L) and line/curve objects (f, g, h, i, j, k, l, m, n, p, q, r, s, t, a, b, d, e, f_1, g_1, h_1, i_1, j_1, k_1, l_1), along with text labels text1 and text2. ④ Suitable for high school or early college mathematics courses focusing on geometric construction exercises, and useful for teachers demonstrating the integration of classical construction with dynamic geometry software to help students understand dependencies and logical order among geometric objects.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Line(C, f) → g
  3. Segment(A, C) → h
  4. Line(B, h) → i
  5. Intersect(g, i) → D
  6. Segment(C, D) → j
  7. Segment(B, D) → k
  8. Segment(C, B) → l
  9. Segment(A, D) → m
  10. Segment(E, F) → n
  11. Circle(E, F) → c
  12. Point(c) → G
  13. Segment(E, G) → p
  14. Line(F, p) → q
  15. Line(G, n) → r
  16. Intersect(q, r) → H
  17. Segment(F, H) → s
  18. Segment(H, G) → t
  19. Segment(F, G) → a
  20. Segment(E, H) → b
  21. Segment(I, J) → d
  22. OrthogonalLine(I, d) → e
  23. OrthogonalLine(J, d) → f_1
  24. Point(e) → K
  25. Line(K, d) → g_1
  26. Intersect(f_1, g_1) → L
  27. Segment(I, J) → h_1
  28. Segment(J, L) → i_1
  29. Segment(L, K) → j_1
  30. Segment(K, I) → k_1
  31. Segment(I, L) → l_1
  32. Segment(J, K) → m_1
  33. Segment(M, N) → n_1
  34. Circle(M, N) → c_1
  35. OrthogonalLine(M, n_1) → p_1
  36. OrthogonalLine(N, n_1) → q_1
  37. Intersect(c_1, p_1, 1) → O
  38. Line(O, n_1) → r_1
  39. Intersect(q_1, r_1) → P
  40. Segment(N, P) → s_1
  41. Segment(P, O) → t_1
  42. Segment(O, M) → a_1
  43. Segment(M, P) → b_1
  44. Segment(N, O) → d_1
  45. Circle(Q, R) → e_1
  46. Point(e_1) → S
  47. Point(e_1) → T
  48. Point(e_1) → U
  49. Segment(T, S) → f_2
  50. Segment(S, R) → g_2
  51. Segment(R, U) → h_2
  52. Segment(U, T) → i_2
  53. Segment(T, R) → j_2
  54. Segment(S, U) → k_2

AI Drawing Prompt Reference

Draw a comprehensive geometric construction: start with a segment between two points, then draw a line through a third point parallel to this segment. Connect the first and third points, and draw a line through the second point parallel to this new segment; the intersection of these two parallel lines gives a point, which is connected to the three vertices to form a triangle. Next, draw a circle with two given points as center and radius, pick a point on the circle, connect it to the center, then through an external point draw a line parallel to this radius, and through that external point draw another line parallel to the original radius; their intersection is connected to form a second triangle. Then, take two parallel lines, draw perpendiculars through their endpoints, pick a point on one perpendicular, draw a line through it parallel to the original lines to meet the other perpendicular, and connect the four points to form a quadrilateral. Finally, draw another circle, construct a perpendicular through the center to the diameter, intersect it with the circle, connect this point to the diameter endpoints to form a triangle, and place three arbitrary points on a separate circle.
The prompt above was compiled from the original drawing commands and is for reference only.