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双心四边形的探索

双心四边形的探索
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2020-05-02 21:37
圆多边形角几何作图
本例通过 GeoGebra 动态演示双心四边形的构造与性质，即同时具外接圆与内切圆的四边形。利用滑块与几何约束，实时展示四边形边长、内角与两圆半径、位置的相互关系。核心数学概念包括外接圆、内切圆、相似三角形、全等三角形及几何变换。构造摘要中涉及的 GeoGebra 命令包括 Circle、Intersect、Polygon、Tangent、Distance、ClosestPoint 等，涉及角度测量、向量运算与多边形绘制。适合高中几何拓展、数学竞赛训练或对圆锥曲线性质感兴趣的学习者。

GeoGebra 命令及参数

  1. ConstructionStep() → ncs
  2. Circle(O1, r1) → ci1
  3. If(ncs ≤ 41, 25°, 0°) → β1
  4. Segment(A1, C1) → dg11
  5. Segment(B1, Intersect[ci1, OrthogonalLine[B1, dg11], 1]) → dg21
  6. Intersect(ci1, OrthogonalLine[B1, dg11], 2) → D1
  7. Intersect(dg11, dg21) → H1
  8. Angle(A1, H1, D1) → α11
  9. Angle(C1, H1, B1) → α21
  10. Polygon(A1, B1, C1, D1) → qu1, a1, b1, c1, d1
  11. Tangent(A1, ci1) → tA1,
  12. Tangent(B1, ci1) → tB1,
  13. Intersect(tA1, tB1) → A2
  14. Tangent(C1, ci1) → tC1,
  15. Intersect(tB1, tC1) → B2
  16. Tangent(D1, ci1) → tD1,
  17. Intersect(tC1, tD1) → C2
  18. Intersect(tD1, tA1) → D2
  19. Polygon(A2, B2, C2, D2) → qu2, a2, b2, c2, d2
  20. Intersect(LineBisector[Segment[A2, B2]], LineBisector[Segment[B2, C2]]) → O2
  21. Distance(O2, A2) → r2
  22. Circle(O2, r2) → ci2
  23. Segment(A2, C2) → dg12
  24. Segment(B2, D2) → dg22
  25. Intersect(dg12, dg22) → H2
  26. Angle(A2, H2, D2) → α12
  27. Angle(C2, H2, B2) → α22
  28. Tangent(A2, ci2) → tA2, g
  29. Tangent(B2, ci2) → tB2,
  30. Intersect(Tangent[A2, ci2], Tangent[B2, ci2]) → A3
  31. Tangent(C2, ci2) → tC2,
  32. Intersect(Tangent[B2, ci2], Tangent[C2, ci2]) → B3
  33. Tangent(D2, ci2) → tD2, h
  34. Intersect(Tangent[C2, ci2], Tangent[D2, ci2]) → C3
  35. Intersect(Tangent[D2, ci2], Tangent[A2, ci2]) → D3
  36. Polygon(A3, B3, C3, D3) → qu3, a3, b3, c3, d3
  37. Intersect(LineBisector[Segment[A3, B3]], LineBisector[Segment[B3, C3]]) → O3
  38. Distance(O3, A3) → r3
  39. Circle(O3, r3) → ci3
  40. ClosestPoint(ci3, D3) → D3'
  41. Tangent(A3, ci3) → tA3,
  42. Tangent(B3, ci3) → tB3,
  43. Tangent(C3, ci3) → tC3,
  44. Tangent(D3, ci3) → tD3, f
  45. Vector(D3, D3') → vD3
  46. Intersect(Tangent[A3, ci3], Tangent[B3, ci3]) → A4
  47. Intersect(Tangent[B3, ci3], Tangent[C3, ci3]) → B4
  48. Intersect(Tangent[C3, ci3], Tangent[D3, ci3]) → C4
  49. Intersect(Tangent[D3, ci3], Tangent[A3, ci3]) → D4
  50. Polygon(A4, B4, C4, D4) → qu4, a4, b4, c4, d4
  51. Intersect(LineBisector[Segment[A4, B4]], LineBisector[Segment[B4, C4]]) → O4
  52. Distance(O4, A4) → r4
  53. Circle(O4, r4) → ci4
  54. ClosestPoint(ci4, D4) → D4'
  55. Vector(D4, D4') → vD4
  56. Circle(Midpoint[D4, D4'], 0.35) → cd4