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Midpoint Coordinates Action: Vectors, Transformations & Parametric Curves

Midpoint Coordinates Action: Vectors, Transformations & Parametric Curves
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2020-08-24 19:25
VectorCoordinate GeometryTransformationParametric Equation
This example demonstrates a dynamic construction driven by midpoint coordinates. It shows how vectors, dilations, rotations, and translations affect the locus of midpoints, linking analytic geometry, vector algebra, similarity transformations, trigonometric functions, and parametric equations. The construction uses GeoGebra commands Midpoint, Dilate, Segment, Rotate, Intersect, Vector, and Translate, manipulating points such as A, B, C, A', C', D, E and auxiliary lines and conics. It is suitable for high‑school analytic geometry and vector/transformation lessons, as well as university‑level geometric modeling or advanced GeoGebra workshops.

GeoGebra Commands & Parameters

  1. Midpoint(A, B) → C
  2. Dilate(A, If[0 < a ≤ 1, 1 - a, 0], C) → A'
  3. Dilate(C, If[1 < a ≤ 2, 2 - a, 0], B) → C'
  4. Segment(A, A') → j
  5. Segment(C, C') → k
  6. Rotate(A, If[2 < a ≤ 3, ((a - 2) * π), π], C) → A'_1
  7. Rotate(A', If[2 < a ≤ 3, ((a - 2) * π), π], C) → A''
  8. Segment(A'_1, A'') → j'
  9. Intersect(i, l) → D
  10. Intersect(f, i) → E
  11. Dilate(E, If[6 < a ≤ 7, 7 - a, 0], D) → E'
  12. Segment(E, E') → m
  13. Dilate(D, If[7 < a ≤ 8, 8 - a, 0], B) → D'
  14. Segment(D, D') → n
  15. Rotate(E, If[8 < a ≤ 9, ((a - 8) * π), π], D) → E'_1
  16. Rotate(E', If[8 < a ≤ 9, ((a - 8) * π), π], D) → E''
  17. Segment(E'_1, E'') → m'
  18. Dilate(B, If[10 < a ≤ 11, 11 - a, 0], D) → B'
  19. Dilate(E, If[10 < a ≤ 11, 11 - a, 0], D) → E'_2
  20. Vector(E, E'_2) → u
  21. Vector(B, B') → v
  22. Intersect(f, p) → F
  23. Intersect(g, p) → G
  24. Dilate(E, If[12 < a ≤ 13, 13 - a, 0], F) → E'_3
  25. Dilate(B, If[12 < a ≤ 13, 13 - a, 0], G) → B'_1
  26. Vector(E, E'_3) → w
  27. Vector(B, B'_1) → b
  28. Translate(D, w) → D'_1
  29. Vector(E'_3, D'_1) → c
  30. Vector(B'_1, D'_1) → d
  31. Dilate(A, If[6 < a ≤ 7, 7 - a, 0], F) → A'_2
  32. Segment(A, A'_2) → q
  33. Dilate(F, If[7 < a ≤ 8, 8 - a, 0], E) → F'
  34. Segment(F, F') → r
  35. Rotate(A, If[8 < a ≤ 9, ((a - 8) * π), π], F) → A'_3
  36. Rotate(A'_2, If[8 < a ≤ 9, ((a - 8) * π), π], F) → H
  37. Segment(A'_3, H) → q'
  38. Dilate(A, If[10 < a ≤ 11, 11 - a, 0], F) → A'_4
  39. Dilate(E, If[10 < a ≤ 11, 11 - a, 0], F) → E'_4
  40. Vector(A, A'_4) → s
  41. Vector(E, E'_4) → t
  42. Dilate(F, If[12 < a ≤ 13, 13 - a, 0], MIDPOINT) → I
  43. Vector(F, I) → u_1
  44. Translate(A, u_1) → A'_5
  45. Vector(A'_5, I) → v_1
  46. Translate(E, u_1) → E'_5
  47. Vector(E'_5, I) → w_1
  48. Translate(F, u_1) → F'_1