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Parallelogram Geometry Task: Incircle and Trigonometric Functions

Parallelogram Geometry Task: Incircle and Trigonometric Functions
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2016-10-12 03:40
CirclePolygonAngleTrigonometryGeometric Construction
This example demonstrates a comprehensive geometric construction involving a parallelogram, an incircle, trigonometric functions, and linear equations. The core mathematical concepts include properties of parallelograms, incircles and tangents, angle measurement, distance calculation, and trigonometric relationships. Construction highlights: using GeoGebra commands such as Segment, Line, Intersect, OrthogonalLine, Point, Circle, Angle, and Distance, the construction builds multiple points (A, D, B, C, E, F, G, H, I, J, K, etc.), segments, lines, perpendiculars, circles, and angle objects (α, β, γ, δ, ε), with text objects (text1, text2, text3) for labeling. This task is suitable for high school or early college students to deepen their understanding of the connection between geometric figures and trigonometric functions, as well as construction and measurement techniques in dynamic geometry software.

GeoGebra Commands & Parameters

  1. Segment(A, D) → f
  2. Segment(A, B) → g
  3. Line(B, f) → h
  4. Line(D, g) → i
  5. Intersect(h, i) → C
  6. Segment(B, C) → j
  7. Segment(D, C) → k
  8. Segment(E, H_1) → l
  9. OrthogonalLine(E, l) → m
  10. Point(m) → F
  11. Segment(F, E) → n
  12. Line(F, l) → p
  13. Line(H_1, n) → q
  14. Intersect(p, q) → G
  15. Segment(F, G) → r
  16. Segment(H_1, G) → s
  17. Segment(H, K) → t
  18. Circle(H, t) → c
  19. Point(c) → I
  20. Segment(H, I) → a
  21. Line(I, t) → b
  22. Line(K, a) → d
  23. Intersect(b, d) → J
  24. Segment(J, I) → e
  25. Segment(J, K) → f_1
  26. Angle(D, A, B) → α
  27. Angle(A, B, C) → β
  28. Angle(B, C, D) → γ
  29. Angle(C, D, A) → δ
  30. Angle(E, F, G) → ε
  31. Angle(G, H_1, E) → ζ
  32. Angle(H_1, E, F) → η
  33. Angle(F, G, H_1) → θ
  34. Angle(H, I, J) → ι
  35. Angle(I, J, K) → κ
  36. Angle(J, K, H) → λ
  37. Angle(K, H, I) → μ
  38. Distance(A, B) → distanceAB
  39. Distance(B, C) → distanceBC
  40. Distance(C, D) → distanceCD
  41. Distance(D, A) → distanceDA
  42. Distance(F, G) → distanceFG
  43. Distance(G, H_1) → distanceGH1
  44. Distance(H_1, E) → distanceH1E
  45. Distance(E, F) → distanceEF
  46. Distance(I, J) → distanceIJ
  47. Distance(J, K) → distanceJK
  48. Distance(K, H) → distanceKH
  49. Distance(H, I) → distanceHI
  50. Segment(A, C) → g_1
  51. Segment(B, D) → h_1
  52. Segment(F, H_1) → i_1
  53. Segment(G, E) → j_1
  54. Segment(H, J) → k_1
  55. Segment(K, I) → l_1
  56. Intersect(g_1, h_1) → L
  57. Intersect(j_1, i_1) → M
  58. Intersect(l_1, k_1) → Q
  59. Distance(A, L) → distanceAL
  60. Distance(B, L) → distanceBL
  61. Distance(L, D) → distanceLD
  62. Distance(C, L) → distanceCL
  63. Distance(F, M) → distanceFM
  64. Distance(M, G) → distanceMG
  65. Distance(H_1, M) → distanceH1M
  66. Distance(M, E) → distanceME
  67. Distance(H, Q) → distanceHN
  68. Distance(I, Q) → distanceIN
  69. Distance(J, Q) → distanceJN
  70. Distance(K, Q) → distanceKN
  71. Distance(E, G) → distanceEG
  72. Distance(F, H_1) → distanceFH1
  73. Angle(J, Q, I) → AngleJQI
  74. Angle(I, K, H) → ν
  75. Angle(J, K, I) → ξ
  76. Angle(H, I, Q) → ο
  77. Angle(K, I, J) → ρ
  78. Angle(I, J, H) → σ
  79. Angle(H, J, K) → τ
  80. Angle(J, H, I) → υ
  81. Angle(K, H, J) → ϕ