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Parallel Lines Proportionality Theorem

Parallel Lines Proportionality Theorem
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2018-08-14 00:47
AngleTransformationGeometric Construction
This example demonstrates the proportionality of parallel lines in geometry. Key mathematical concepts include similar triangles, proportional segments, and geometric construction. The construction uses GeoGebra commands like Line, Slope, PointIn, and others to create parallel lines and verify proportions. Suitable for middle school or high school students for geometry learning and classroom activities.

GeoGebra Commands & Parameters

  1. Line(C, B_1) → a
  2. Slope(a) → b
  3. PointIn(d) → A
  4. Segment(A, C) → i
  5. Point(i) → B
  6. Distance(A, B) → n
  7. Circle(A, 1) → f
  8. Line(A, a) → e
  9. Intersect(f, e, 2) → D_1
  10. Intersect(f, e, 1) → E_1
  11. Ray(A, D_1) → g
  12. Ray(A, E_1) → h
  13. Point(g) → D
  14. Ray(C, B_1) → j_1
  15. Point(j_1) → F
  16. Circle(C, 1) → k
  17. Intersect(k, a, 1) → I
  18. Line(B, a) → l
  19. Segment(D, F) → m
  20. Intersect(l, m) → E
  21. Circle(B, 1) → p
  22. Intersect(p, l, 2) → K
  23. Intersect(p, i, 1) → L
  24. CircleSector(B, K, L) → q
  25. Dilate(B, If[0 < j ≤ 1, 1 - j, 0], A) → H'
  26. Vector(B, H') → u
  27. Translate(q, u) → q'
  28. Rotate(q', If[1 < j ≤ 2, (((j - 1) * 180))°, 180°], A) → q''
  29. Dilate(B, If[2 < j ≤ 3, 3 - j, 0], C) → H'_1
  30. Vector(B, H'_1) → v
  31. Translate(q, v) → q'_1
  32. Segment(B, A) → r
  33. Segment(B, C) → s
  34. Segment(D, E) → t
  35. Segment(E, F) → a_1
  36. Ray(C, I) → b_1
  37. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], A) → F'
  38. Vector(D, F') → w
  39. Translate(D, w) → F'_1
  40. Translate(E, w) → E'
  41. Segment(F'_1, E') → t'
  42. Translate(E, w) → J'_1
  43. Translate(F, w) → G'_1
  44. Segment(J'_1, G'_1) → c_1
  45. Polygon(B, A, E') → poly1, j', h_1, c_2
  46. Dilate(B, o - 1.5, A) → H'_2
  47. Dilate(A, o - 1.5, A) → C'
  48. Dilate(E', o - 1.5, A) → J''
  49. Dilate(A, o - 0.5, A) → C'_1
  50. Dilate(E', o - 0.5, A) → J''_1
  51. Dilate(A, ((o - 0.5) * e_1), A) → C'_2
  52. Dilate(E', ((o - 0.5) * e_1), A) → J''_2
  53. Dilate(B, If[1.5 < o ≤ 2.5, 2.5 - o, 0], C) → H'_3
  54. Dilate(E', If[1.5 < o ≤ 2.5, 2.5 - o, 0], G'_1) → J''_3
  55. Polygon(H'_3, A, J''_3) → poly2, j''_3, h'_3, c_3
  56. Dilate(E', If[1.5 < o ≤ 2.5, 2.5 - o, 0], G'_1) → E''