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Goat Clipper (Hyperclassic Problem): Constructing an Optimal Path via Circle/Con

Goat Clipper (Hyperclassic Problem): Constructing an Optimal Path via Circle/Con
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2021-01-16 19:20
CircleAngleVectorTransformationGeometric Construction
This GeoGebra example demonstrates how the “Goat Clipper” hyperclassic problem can be visualized and solved through synthetic geometry construction in the plane. It shows how, given walls/constraints and key points, one can determine the crucial optimal connecting path by exploiting the geometric locus behind a difference-and-distance condition—often linked to conic behavior such as a hyperbola-type “difference of distances” property. The core concepts include circle and arc geometry, angles, vectors, distance measurements, and transformations (mirror/rotation) that convert the constrained problem into an equivalent construction. In the construction process, it uses Segment and Point to define the geometric skeleton; Circle, CircleArc, and CircleSector to create the required circles and arcs; Midpoint and LineBisector for symmetry and angle-bisector structures; Mirror and Rotate to generate transformed copies and Intersect to compute key intersection points; Angle, Ray, Distance, and Vector to measure direction/angles and compute required distance relations; Root, If, and Sum for selecting or computing the relevant solution components; and Polygon/Arc to present the final feasible region or critical curve segment clearly. Suitable for advanced high-school contest preparation and for instructional demonstrations in geometry construction and conic-locus reasoning.

GeoGebra Commands & Parameters

  1. Segment(G, R) → gr
  2. Point(gr) → A
  3. CircleSector(G, L_21, L_11) → zone3
  4. CircleSector(E, L_31, L_21) → zone5
  5. CircleSector(A, L_11, L_12) → zone1
  6. CircleSector(R, L_12, L_22) → zone2
  7. Midpoint(L_31, L_32) → L_4
  8. CircleSector(D, L_22, L_4) → zone4
  9. Point(l1) → B
  10. Segment(G, A) → ga
  11. If(y(B) ≤ y(R), 1, 0) → A1
  12. If(y(B) > y(R) ∧ x(B) ≥ x(R), 2, 0) → A2
  13. If(y(B) ≥ y(D) ∧ x(B) < x(D) ∧ x(B) > x(L_4), 4, 0) → A4
  14. If(y(B) ≥ y(E) ∧ x(B) > x(E) ∧ x(B) < x(L_4), 5, 0) → A5
  15. If(y(B) > y(R) ∧ x(B) ≤ x(G), 3, 0) → A3
  16. Sum(A1:A5) → A7
  17. Object("B" + A7) → B7
  18. Object("C" + A7) → C7
  19. Midpoint(B_{2}, B) → F
  20. Segment(B_{2}, F) → bf
  21. Root(f, -1.5, 1.5) → H
  22. Rotate(B, x(H), B_{2}) → C'
  23. Angle(B, B_{2}, C') → δ
  24. LineBisector(bf) → a
  25. Ray(B_{2}, C') → b
  26. Intersect(a, b) → I
  27. CircleArc(I, B_{2}, F) → c
  28. Mirror(I, F) → I'
  29. Mirror(B_{2}, F) → B'
  30. Mirror(F, F) → F'
  31. Arc(Mirror[Circle[I, B_{2}], F], B', F') → c'
  32. Segment(A, G) → ag
  33. Segment(E, G) → eg
  34. Segment(A, R) → ar
  35. Segment(R, D) → rd
  36. Ray(B_{2}, B) → d
  37. Circle(B_{2}, C7) → e
  38. Intersect(e, d) → K,
  39. Segment(B_{2}, K) → bk
  40. If((2 * bf) ≥ C7, K, B) → J
  41. Rotate(P, 320°, T) → P'
  42. Angle(P, T, P') → α
  43. Segment(T, P') → h
  44. Polygon(P, E, G, R, D) → poly1, p, e_1, g_1, r, d_1
  45. Segment(P, T) → i
  46. Segment(G, A) → j
  47. If(4.45 < j ∧ j < 4.55 ∧ 10.45 < lc ∧ lc < 10.55, true, false) → bs
  48. Segment(E, G) → k
  49. Segment(E, P) → l
  50. Segment(T, D) → m
  51. Segment(D, R) → n
  52. Segment(P', T) → q
  53. Segment(G', R') → s
  54. Segment(R, R') → t
  55. Segment(G, G') → a_1
  56. Midpoint(G, R) → Z
  57. Circle(A, lc) → c_1
  58. CircleArc(A, L_11, L_12) → f_1
  59. CircleArc(R, L_12, L_22) → h_1
  60. CircleArc(G, L_21, L_11) → k_1
  61. CircleArc(D, L_22, L_4) → p_1
  62. CircleArc(E, L_31, L_21) → q_1
  63. Circle(A, lc) → r_1
  64. Segment(A, R) → i_1
  65. Angle(M, A, B) → β
  66. Segment(R, D) → j_1
  67. Segment(A, G) → l_1
  68. Segment(G, E) → m_1
  69. Distance(G, R) → dis
  70. Distance(A, R) → disa
  71. Distance(A, G) → disb
  72. Midpoint(C, N) → D_1
  73. Midpoint(C, O) → E_1
  74. Vector(D_1, N) → u
  75. Vector(D_1, C) → v
  76. Vector(E_1, C) → w
  77. Vector(E_1, O) → u_1
  78. Midpoint(G_1, F_1) → H_1
  79. Vector(H_1, G_1) → v_1
  80. Vector(H_1, F_1) → w_1

AI Drawing Prompt Reference

Create a planar diagram illustrating the “Goat Clipper” hyperclassic geometry construction. Start with given key points and draw the main connecting segments that represent walls/constraints. Build several circular sector regions using circles and arc segments centered at the specified points, forming overlapping feasible zones. Use a midpoint-derived line (such as a perpendicular/angle-bisector structure) together with a ray to compute a crucial intersection point. Construct a circular arc through that intersection and the midpoint, then generate corresponding elements by mirroring the point and the arc about the relevant center/line, along with a rotation to obtain the rotated counterpart. Combine the resulting rays, segments, and arcs into a final polygon/arc boundary that highlights the constrained, conic-locus–like path reasoning, including the key distances and an angle measurement showing the directional constraint.
The prompt above was compiled from the original drawing commands and is for reference only.