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Cats on a Ladder: Ellipse and Hyperbola Geometric Demonstration

Cats on a Ladder: Ellipse and Hyperbola Geometric Demonstration
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2015-07-04 06:09
TransformationGeometric ConstructionLocus
This GeoGebra example demonstrates the generation of ellipses and hyperbolas through a ladder (segment) with 'cats' (points) placed on it, using mirror and midpoint constructions. The core mathematical concepts include ellipses, hyperbolas, similar triangles, geometric distribution, and integer equations. The construction employs GeoGebra commands such as Intersect, Midpoint, Mirror, Segment, Ellipse, and Polygon, creating multiple point objects (A, B, C, D) and series of points (A1-A7, B1-B7, etc.), with a button (button2) to control dynamic changes. This is suitable for high school or early college students to intuitively understand conic sections, geometric transformations, and locus relationships, and can serve as material for math lab activities or inquiry-based learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Midpoint(C, D) → A1
  3. Midpoint(A1, C) → A2
  4. Midpoint(A2, C) → A3
  5. Midpoint(A2, B) → A4
  6. Midpoint(B, D) → A5
  7. Midpoint(A5, D) → A6
  8. Midpoint(A5, B) → A7
  9. Mirror(A1, yAxis) → C1
  10. Mirror(A2, yAxis) → C2
  11. Mirror(A3, yAxis) → C3
  12. Mirror(A4, yAxis) → C4
  13. Mirror(A5, yAxis) → C5
  14. Mirror(A6, yAxis) → C6
  15. Mirror(A7, yAxis) → C7
  16. Mirror(B1, yAxis) → D1
  17. Mirror(B2, yAxis) → D2
  18. Mirror(B3, yAxis) → D3
  19. Mirror(B4, yAxis) → D4
  20. Mirror(B5, yAxis) → D5
  21. Mirror(B6, yAxis) → D6
  22. Mirror(B7, yAxis) → D7
  23. Mirror(D, yAxis) → D'
  24. Segment(C, D) → e
  25. Ellipse(E1, F1, G1) → H1
  26. Polygon(F1, G1, 3) → I1, f, h, i, F
  27. Polygon(F2, G2, 3) → I2, m, n, p, H
  28. Polygon(F3, G3, 3) → I3, q, s, a_1, I
  29. Polygon(F4, G4, 3) → I4, b_1, c_1, d_1, J
  30. Polygon(F5, G5, 3) → I5, e_1, f_1, g_1, K
  31. Polygon(F6, G6, 3) → I6, h_1, i_1, j_1, L
  32. Polygon(F7, G7, 3) → I7, k_1, l_1, m_1, M
  33. Polygon(F8, G8, 3) → I8, n_1, p_1, q_1, N
  34. Polygon(F9, G9, 3) → I9, r_1, s_1, t_1, O
  35. Polygon(G1, E1, 3) → N1, j, k, l, G
  36. Polygon(G2, E2, 3) → N2, a_2, b_2, c_2, P
  37. Polygon(G3, E3, 3) → N3, d_2, e_2, f_2, Q
  38. Polygon(G4, E4, 3) → N4, g_2, h_2, i_2, R
  39. Polygon(G5, E5, 3) → N5, j_2, k_2, l_2, S
  40. Polygon(G6, E6, 3) → N6, m_2, n_2, p_2, T
  41. Polygon(G7, E7, 3) → N7, q_2, r_2, s_2, U
  42. Polygon(G8, E8, 3) → N8, t_2, a_3, b_3, V
  43. Polygon(G9, E9, 3) → N9, c_3, d_3, e_3, W
  44. Ellipse(E2, F2, G2) → H2
  45. Ellipse(E3, F3, G3) → H3
  46. Ellipse(E4, F4, G4) → H4
  47. Ellipse(E5, F5, G5) → H5
  48. Ellipse(E6, F6, G6) → H6
  49. Ellipse(E7, F7, G7) → H7
  50. Ellipse(E8, F8, G8) → H8
  51. Ellipse(E9, F9, G9) → H9