积极 AI Logo 积极 AI

直角三角形中的正方形

直角三角形中的正方形
几何画板及绘图内容加载中，请稍等……
2016-03-05 16:38
解析几何变换最值优化几何作图
①本范例演示在直角三角形内部构造一个正方形的方法；②涉及的核心数学概念包括几何作图、坐标几何、变换和优化；③作图要点——使用了Polygon（多边形）、Intersect（交点）、Segment（线段）、Point（点）、Min（最小值）、If（条件）、Max（最大值）、Rotate（旋转）等GeoGebra命令和对象；④适合高中数学教学或几何爱好者探索图形性质。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Intersect(Line[C, C + (1, 1)], c) → D
  3. Polygon(C, E, D, F) → poly2, c_1, e, d, f
  4. Polygon(A, D, E) → poly3, e_1, a_1, d_1
  5. Polygon(D, B, F) → poly4, f_1, d_2, b_1
  6. Segment(G, G + (9, 0)) → g
  7. Point(g) → H
  8. Min((x(H) - x(G)) / (g / 3), 1) → t1
  9. If(x(H) < x(G) + 3, 0, If[x(H) < x(G) + 6, (x(H) - x(G) - g / 3) / (g / 3), 1]) → t2
  10. Max((x(H) - x(G) - (2 * g / 3)) / (g / 3), 0) → t3
  11. Rotate(Polygon[Translate[C, Vector[(t1 * Vector[C, I])]], Translate[E, Vector[(t1 * Vector[C, I])]], Translate[D, Vector[(t1 * Vector[C, I])]], Translate[F, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly5
  12. Rotate(Polygon[Translate[A, Vector[(t1 * Vector[C, I])]], Translate[D, Vector[(t1 * Vector[C, I])]], Translate[E, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly6
  13. Rotate(Polygon[Translate[D, Vector[(t1 * Vector[C, I])]], Translate[B, Vector[(t1 * Vector[C, I])]], Translate[F, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly7
  14. Rotate(Translate[D, Vector[(t1 * Vector[C, I])]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → J
  15. Rotate(Translate[E, Vector[(t1 * Vector[C, I])]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → K
  16. Rotate(Translate[F, Vector[(t1 * Vector[C, I])]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → L
  17. Rotate(Polygon[Translate[C, Vector[(t2 * Vector[C, M])]], Translate[E, Vector[(t2 * Vector[C, M])]], Translate[D, Vector[(t2 * Vector[C, M])]], Translate[F, Vector[(t2 * Vector[C, M])]]], 0°, Translate[C, Vector[(t2 * Vector[C, M])]]) → poly8
  18. Rotate(Polygon[Translate[D, Vector[(t2 * Vector[C, M])]], Translate[B, Vector[(t2 * Vector[C, M])]], Translate[F, Vector[(t2 * Vector[C, M])]]], 0°, Translate[C, Vector[(t2 * Vector[C, M])]]) → poly9
  19. Rotate(Translate[poly7, Vector[(t2 * Vector[J, N])]], 0°, Translate[J, Vector[(t2 * Vector[J, N])]]) → poly10
  20. Rotate(Translate[poly6, Vector[(t3 * Vector[B, N])]], ((t3 * 90))°, Translate[B, Vector[(t3 * Vector[B, N])]]) → poly11
  21. Rotate(Translate[poly5, Vector[(t3 * Vector[B, N])]], ((t3 * 90))°, Translate[B, Vector[(t3 * Vector[B, N])]]) → poly12
  22. Rotate(Polygon[Translate[A, Vector[(t3 * Vector[D, N])]], Translate[D, Vector[(t3 * Vector[D, N])]], Translate[E, Vector[(t3 * Vector[D, N])]]], ((t3 * 90))°, Translate[D, Vector[(t3 * Vector[D, N])]]) → poly13
  23. Polygon(C, E, D, F) → poly14, c_2, e_2, d_3, f_2
  24. Polygon(A, D, E) → poly15, e_3, a_2, d_4
  25. Polygon(D, B, F) → poly16, f_3, d_5, b_2
  26. Rotate(Polygon[Translate[C, Vector[(t1 * Vector[C, I])]], Translate[E, Vector[(t1 * Vector[C, I])]], Translate[D, Vector[(t1 * Vector[C, I])]], Translate[F, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly17
  27. Rotate(Polygon[Translate[D, Vector[(t1 * Vector[C, I])]], Translate[B, Vector[(t1 * Vector[C, I])]], Translate[F, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly18
  28. Rotate(Polygon[Translate[A, Vector[(t1 * Vector[C, I])]], Translate[D, Vector[(t1 * Vector[C, I])]], Translate[E, Vector[(t1 * Vector[C, I])]]], ((t1 * 180))°, Translate[C, Vector[(t1 * Vector[C, I])]]) → poly19
  29. Rotate(Polygon[Translate[C, Vector[(t2 * Vector[C, M])]], Translate[E, Vector[(t2 * Vector[C, M])]], Translate[D, Vector[(t2 * Vector[C, M])]], Translate[F, Vector[(t2 * Vector[C, M])]]], 0°, Translate[C, Vector[(t2 * Vector[C, M])]]) → poly20
  30. Rotate(Polygon[Translate[D, Vector[(t2 * Vector[C, M])]], Translate[B, Vector[(t2 * Vector[C, M])]], Translate[F, Vector[(t2 * Vector[C, M])]]], 0°, Translate[C, Vector[(t2 * Vector[C, M])]]) → poly21
  31. Rotate(Translate[poly7, Vector[(t2 * Vector[J, N])]], 0°, Translate[J, Vector[(t2 * Vector[J, N])]]) → poly22
  32. Rotate(Polygon[Translate[A, Vector[(t3 * Vector[D, N])]], Translate[D, Vector[(t3 * Vector[D, N])]], Translate[E, Vector[(t3 * Vector[D, N])]]], ((t3 * 90))°, Translate[D, Vector[(t3 * Vector[D, N])]]) → poly23
  33. Rotate(Translate[poly6, Vector[(t3 * Vector[B, N])]], ((t3 * 90))°, Translate[B, Vector[(t3 * Vector[B, N])]]) → poly24
  34. Rotate(Translate[poly5, Vector[(t3 * Vector[B, N])]], ((t3 * 90))°, Translate[B, Vector[(t3 * Vector[B, N])]]) → poly25
  35. Segment(B, I) → h
  36. Segment(A, I) → i
  37. Segment(M, N) → j
  38. Segment(N, Q) → k
  39. Segment(P, Q) → l
  40. Segment(M, P) → m
  41. Segment(R, N) → n
  42. Segment(R, S) → p
  43. Segment(S, Q) → q