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Proof Without Words by Steve Phelps

Proof Without Words by Steve Phelps
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2019-01-14 04:55
TrigonometryCoordinate GeometryCalculusOptimizationGeometric Construction
This example visually demonstrates the Intermediate Value Theorem for continuous functions through dynamic geometry. Key math concepts include continuity, slope, and triangle properties. GeoGebra commands like Point, Mirror, Semicircle, Intersect, CircleArc, Line, AngularBisector, OrthogonalLine, Circle, Polygon, Midpoint, Rotate, Angle, If are used to construct the geometric model. Suitable for high school students and math enthusiasts to visualize abstract concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Mirror(A, yAxis) → A'
  3. Semicircle(A', A) → c
  4. Intersect(c, yAxis, 1) → B
  5. Intersect(xAxis, yAxis) → C
  6. CircleArc(C, B, A') → d
  7. Point(d) → D
  8. Line(A', D) → f
  9. Line(D, A) → g
  10. Line(A, A') → h
  11. AngularBisector(f, h) → i, j
  12. AngularBisector(h, g) → k, l
  13. Intersect(j, k) → E
  14. Intersect(i, l) → F
  15. Intersect(i, k) → G
  16. OrthogonalLine(E, g) → m
  17. OrthogonalLine(F, f) → n
  18. OrthogonalLine(G, h) → p
  19. Intersect(h, p) → H
  20. Intersect(g, m) → I
  21. Intersect(f, n) → J
  22. Circle(F, J) → e
  23. Circle(E, I) → q
  24. Circle(G, H) → r
  25. Polygon(A', D, 4) → poly1, s, t, a, b, K, L
  26. Polygon(D, A, 4) → poly2, f_1, g_1, h_1, i_1, M, N
  27. Polygon(A, A', 4) → poly3, j_1, k_1, l_1, m_1, O, P
  28. Midpoint(L, D) → Q
  29. Midpoint(D, M) → R
  30. Mirror(J, Q) → J'
  31. Mirror(I, R) → I'
  32. Rotate(J, 90°, Q) → J'_1
  33. Mirror(J'_1, Q) → S
  34. Polygon(S, J, 4) → poly4, n_1, p_1, q_1, r_1, T, U
  35. Rotate(I, 90°, R) → II
  36. Polygon(I, II, 4) → poly5, s_1, t_1, a_1, b_1, V, W
  37. Midpoint(A', P) → Z
  38. Polygon(A', S, J) → t1, j_2, a', s_2
  39. Polygon(S, L, J') → t2, j', s_3, l_2
  40. Polygon(J', K, J'_1) → t3, j'_1, c_1, k_2
  41. Polygon(J'_1, D, J) → t4, j_3, d_1, d_2
  42. Polygon(W, I', N) → t5, n_2, w, i'
  43. Polygon(W, D, I) → t6, i_2, w_1, d_3
  44. Polygon(I, A, II) → t7, i'_1, i_3, a_2
  45. Polygon(II, I', M) → t8, m_2, e_1, f_2
  46. Polygon(II, R, I') → t9, g_2, h_2, r_2
  47. Polygon(II, R, I) → t10, i_4, p_2, r_3
  48. Polygon(I, R, W) → t11, w_2, i_5, r_4
  49. Polygon(R, W, I') → t12, q_2, r_5, w_3
  50. Angle(A, A', D) → α
  51. Angle(D, A, A') → β
  52. If(0 < o ≤ 1, Dilate[S, 1 - o, O], O) → A_1
  53. If(0 < o ≤ 1, Dilate[U, 1 - o, A'], A') → B_1
  54. If(0 < o ≤ 1, Dilate[T, 1 - o, A], A) → C_1
  55. If(0 < o ≤ 1, Dilate[J, 1 - o, P], P) → D_1
  56. If(0 < o ≤ 1, Rotate[Polygon[Translate[A', A_1 - S], Translate[S, A_1 - S], Translate[J, A_1 - S]], (o * β), A_1], Rotate[Polygon[Translate[A', A_1 - S], Translate[S, A_1 - S], Translate[J, A_1 - S]], β, A_1]) → t13
  57. If(0 < o ≤ 1, Rotate[Polygon[Translate[S, B_1 - U], Translate[L, B_1 - U], Translate[J', B_1 - U]], (o * β), B_1], Rotate[Polygon[Translate[S, B_1 - U], Translate[L, B_1 - U], Translate[J', B_1 - U]], β, A']) → t14
  58. If(0 < o ≤ 1, Rotate[Polygon[Translate[J', C_1 - T], Translate[K, C_1 - T], Translate[J'_1, C_1 - T]], (o * β), C_1], Rotate[Polygon[Translate[J', C_1 - T], Translate[K, C_1 - T], Translate[J'_1, C_1 - T]], β, A]) → t15
  59. If(0 < o ≤ 1, Rotate[Polygon[Translate[J'_1, D_1 - J], Translate[D, D_1 - J], Translate[J, D_1 - J]], (o * β), D_1], Rotate[Polygon[Translate[J'_1, D_1 - J], Translate[D, D_1 - J], Translate[J, D_1 - J]], β, P]) → t16
  60. If(2 < o ≤ 3, Dilate[I, 3 - o, O], O) → E_1
  61. If(2 < o ≤ 3, Dilate[W, 3 - o, A'], A') → F_1
  62. If(2 < o ≤ 3, Dilate[V, 3 - o, A], A) → G_1
  63. If(2 < o ≤ 3, Dilate[II, 3 - o, P], P) → H_1
  64. If(2 < o ≤ 3, Rotate[Polygon[Translate[W, E_1 - I], Translate[D, E_1 - I], Translate[I, E_1 - I]], ((o - 2) * (β - 90°)), E_1], Rotate[Polygon[Translate[W, E_1 - I], Translate[D, E_1 - I], Translate[I, E_1 - I]], β - 90°, O]) → t17
  65. If(2 < o ≤ 3, Rotate[Polygon[Translate[W, F_1 - W], Translate[I', F_1 - W], Translate[N, F_1 - W]], ((o - 2) * (β - 90°)), F_1], Rotate[Polygon[Translate[W, F_1 - W], Translate[I', F_1 - W], Translate[N, F_1 - W]], β - 90°, A']) → t18
  66. If(2 < o ≤ 3, Rotate[Polygon[Translate[II, G_1 - V], Translate[I', G_1 - V], Translate[M, G_1 - V]], ((o - 2) * (β - 90°)), G_1], Rotate[Polygon[Translate[II, G_1 - V], Translate[I', G_1 - V], Translate[M, G_1 - V]], β - 90°, A]) → t19
  67. If(2 < o ≤ 3, Rotate[Polygon[Translate[I, H_1 - II], Translate[A, H_1 - II], Translate[II, H_1 - II]], ((o - 2) * (β - 90°)), H_1], Rotate[Polygon[Translate[I, H_1 - II], Translate[A, H_1 - II], Translate[II, H_1 - II]], β - 90°, P]) → t20
  68. If(4 < o ≤ 5, Dilate[I, 5 - o, O], O) → I_1
  69. If(4 < o ≤ 5, Dilate[W, 5 - o, A'], A') → J_1
  70. If(4 < o ≤ 5, Dilate[V, 5 - o, A], A) → K_1
  71. If(4 < o ≤ 5, Dilate[II, 5 - o, P], P) → L_1
  72. If(4 < o ≤ 5, Rotate[Polygon[Translate[I, I_1 - I], Translate[R, I_1 - I], Translate[W, I_1 - I]], ((o - 4) * (β - 90°)), I_1], Rotate[Polygon[Translate[I, I_1 - I], Translate[R, I_1 - I], Translate[W, I_1 - I]], β - 90°, O]) → t21
  73. If(4 < o ≤ 5, Rotate[Polygon[Translate[R, J_1 - W], Translate[W, J_1 - W], Translate[I', J_1 - W]], ((o - 4) * (β - 90°)), J_1], Rotate[Polygon[Translate[R, J_1 - W], Translate[W, J_1 - W], Translate[I', J_1 - W]], β - 90°, A']) → t22
  74. If(4 < o ≤ 5, Rotate[Polygon[Translate[II, K_1 - V], Translate[R, K_1 - V], Translate[I', K_1 - V]], ((o - 4) * (β - 90°)), K_1], Rotate[Polygon[Translate[II, K_1 - V], Translate[R, K_1 - V], Translate[I', K_1 - V]], β - 90°, A]) → t23
  75. If(4 < o ≤ 5, Rotate[Polygon[Translate[II, L_1 - II], Translate[R, L_1 - II], Translate[I, L_1 - II]], ((o - 4) * (β - 90°)), L_1], Rotate[Polygon[Translate[II, L_1 - II], Translate[R, L_1 - II], Translate[I, L_1 - II]], β - 90°, P]) → t24
  76. If(6 < o ≤ 7, Dilate[Q, 7 - o, Z], Z) → M_1
  77. Angle(A', S, J) → δ
  78. Angle(W, I, D) → ε
  79. If(6 < o ≤ 7, Rotate[Translate[poly4, M_1 - Q], ((o - 6) * β), M_1], Rotate[Translate[poly4, M_1 - Q], β, Z]) → q1
  80. Angle(A', D, A) → ζ

AI Drawing Prompt Reference

Create a dynamic geometric model demonstrating the Intermediate Value Theorem for continuous functions. Include a semicircle and two intersecting arcs, showing continuity, slope, and triangle properties. Start with horizontal and vertical lines, mirror the semicircle across the vertical line, construct arcs and lines, draw angle bisectors and perpendiculars, and finally form multiple polygons and circles to highlight geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.