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Proof Without Words: Quadratic Equation Demonstration

Proof Without Words: Quadratic Equation Demonstration
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2015-12-13 01:30
Coordinate GeometryGeometric Construction
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This example visually demonstrates the relationship between quadratic equations and geometric figures through dynamic geometry. Key math concepts include quadratic equations, means inequality, natural numbers, and matrix operations. GeoGebra commands like Point, Mirror, Semicircle, Intersect, CircleArc, Polygon, OrthogonalLine, Dilate, Vector, and Translate are used to construct the geometric representation. Suitable for high school and above students to understand abstract mathematical concepts intuitively.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(yAxis) → B
  3. Mirror(A, B) → A'
  4. Semicircle(A, A') → c
  5. Intersect(c, yAxis, 1) → C
  6. CircleArc(B, C, A) → d
  7. Point(d) → D
  8. Polygon(A, D, 4) → poly1, a, b, e, f, E, F
  9. Polygon(D, A', 4) → poly2, g, h, i, j, G, H
  10. Polygon(A', A, 4) → poly3, k, l, m, n, I, J
  11. OrthogonalLine(A', k) → p
  12. OrthogonalLine(D, p) → q
  13. Intersect(p, q) → K
  14. Intersect(p, i) → L
  15. Intersect(q, h) → M
  16. Polygon(D, K, L, H) → poly4, d_1, k_1, l_1, h_1
  17. Polygon(K, L, G, M) → poly5, k_2, l_2, g_1, m_1
  18. Polygon(A', K, M) → poly6, m_2, a', k_3
  19. Polygon(A', K, D) → poly7, d_2, a'_1, k_4
  20. Dilate(K, 1 - o, J) → K'
  21. Dilate(K, 1 - o, A') → K'_1
  22. Dilate(K, 1 - o, I) → K'_2
  23. Dilate(K, 1 - o, A) → K'_3
  24. Vector(K, K'_1) → u
  25. Vector(K, K') → v
  26. Vector(K, K'_2) → w
  27. Vector(K, K'_3) → r
  28. Translate(K, w) → K'_4
  29. Translate(L, w) → L'
  30. Translate(G, w) → G'
  31. Translate(M, w) → M'
  32. Polygon(K'_4, L', G', M') → poly5', k'_4, l', g', m'
  33. Translate(D, v) → D'
  34. Translate(K, v) → K'_5
  35. Translate(L, v) → L'_1
  36. Translate(H, v) → H'
  37. Polygon(D', K'_5, L'_1, H') → poly4', d', k'_5, l'_1, h'
  38. Translate(A', r) → A''
  39. Translate(K, r) → K'_6
  40. Translate(M, r) → M'_1
  41. Polygon(A'', K'_6, M'_1) → poly6', m'_1, a'', k'_6
  42. Translate(A', u) → A''_1
  43. Translate(K, u) → K'_7
  44. Translate(D, u) → D'_1
  45. Polygon(A''_1, K'_7, D'_1) → poly7', d'_1, a''_1, k'_7
  46. Dilate(E, 1 - o, D'_1) → E'
  47. Vector(E, E') → s
  48. Polygon(F, E, D, A) → poly8, f_1, e_1, d_3, a_1
  49. Translate(F, s) → F'
  50. Translate(E, s) → E'_1
  51. Translate(D, s) → D'_2
  52. Translate(A, s) → A'_1
  53. Polygon(F', E'_1, D'_2, A'_1) → poly8', f', e'_1, d'_2, a'_2