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平方根与和：动态几何中的距离与方差探究

平方根与和：动态几何中的距离与方差探究
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2017-05-29 00:25
函数解析几何微积分几何作图函数图像
本范例通过GeoGebra动态几何环境，演示了平方根运算与距离求和之间的数学关系，并融合了方差、反射、切线等概念。核心图形包括多个多边形、圆、线段以及由交点、中点、垂线等构造的辅助元素，通过测量距离（如EG、BC、EB）并计算其平方根或和，直观展示代数表达式与几何量之间的对应。作图要点涉及使用Point、Line、Segment、Polygon、Circle、Intersect、Distance、Midpoint、OrthogonalLine等命令，构建了包含点A、B、C、D、E、F、G、H、I、J、K以及多边形poly1、poly2、poly3的复杂结构，并利用文本对象（TextEG、TextBC等）动态显示计算结果。该范例适合高中或大学低年级学生，用于理解平方根函数的几何意义、距离公式、方差概念以及微分几何中切线与反射的直观联系，也可作为探究性学习的素材。

GeoGebra 命令及参数

  1. Point(yAxis) → A
  2. Line(A, xAxis) → a_1
  3. Point(Circle[B, a]) → C
  4. Segment(B, C) → c_2
  5. Polygon(B, C, 4) → poly1, d, e, f, g, D, E
  6. Line(E, C) → h
  7. Circle(E, b) → k
  8. Intersect(k, h) → F, G
  9. Polygon(E, G, 4) → poly2, i, j, l, m, H, I
  10. Distance(E, G) → distanceEG
  11. Distance(B, C) → distanceBC
  12. Distance(E, B) → distanceEB
  13. Point(Circle[J, a]) → K
  14. Segment(J, K) → p
  15. Polygon(J, K, 4) → poly3, q, r, s, t, L, M
  16. Polygon(M, K, 4) → poly4, b_1, c_1, d_1, e_1, N, O
  17. Distance(J, K) → distanceJK
  18. Distance(J, M) → distanceJM
  19. Distance(M, K) → distanceMK
  20. Distance(K, N) → distanceKN
  21. Midpoint(M, K) → P
  22. Midpoint(K, N) → Q
  23. Midpoint(J, K) → R
  24. Midpoint(J, M) → S
  25. Segment(S, P) → f_1
  26. Segment(P, R) → g_1
  27. Midpoint(M, O) → T
  28. Midpoint(O, N) → U
  29. Segment(T, Q) → h_1
  30. Segment(P, U) → i_1
  31. Intersect(p, q) → V
  32. Intersect(p, q) → W
  33. Intersect(p, q) → Z
  34. Circle(J, 1) → c
  35. Circle(J, 2) → k_1
  36. Intersect(c, q) → A_1,
  37. Intersect(t, poly3) → B_1, C_1
  38. Intersect(k_1, q) → D_1,
  39. Intersect(t, poly3) → E_1, F_1
  40. Intersect(k_1, t) → G_1,
  41. Intersect(c, t) → H_1,
  42. OrthogonalLine(D_1, p) → j_1
  43. OrthogonalLine(A_1, p) → l_1
  44. OrthogonalLine(H_1, t) → m_1
  45. OrthogonalLine(G_1, t) → n_1
  46. Intersect(n_1, b_1) → I_1
  47. Intersect(m_1, b_1) → J_1
  48. OrthogonalLine(J_1, b_1) → p_1
  49. OrthogonalLine(I_1, b_1) → q_1
  50. Intersect(c_1, m_1) → K_1
  51. Intersect(c_1, n_1) → L_1
  52. OrthogonalLine(K_1, c_1) → r_1
  53. OrthogonalLine(L_1, c_1) → s_1
  54. Midpoint(R, K) → M_1
  55. OrthogonalLine(M_1, p) → t_1
  56. Midpoint(S, M) → N_1
  57. OrthogonalLine(N_1, t) → a_2
  58. Intersect(e_1, t_1) → O_1
  59. OrthogonalLine(O_1, e_1) → b_2
  60. Intersect(poly3, poly4) → P_1, Q_1, R_1, S_1, T_1, U_1, V_1, W_1
  61. Intersect(b_1, a_2) → Z_1
  62. OrthogonalLine(Z_1, b_1) → d_2
  63. Intersect(d_1, d_2) → A_2
  64. Intersect(d_1, p_1) → B_2
  65. Intersect(b_2, c_1) → C_2
  66. Segment(N_1, Z_1) → e_2
  67. Segment(Z_1, A_1) → f_2
  68. Segment(H_1, J_1) → g_2
  69. Segment(J_1, M_1) → h_2
  70. Segment(J_1, B_2) → i_2
  71. Segment(Z_1, A_2) → j_2
  72. Segment(O_1, C_2) → k_2
  73. Intersect(r_1, e_1) → D_2
  74. Segment(D_2, K_1) → l_2
  75. Segment(M, O) → m_2
  76. Segment(O, N) → n_2
  77. Segment(N, K) → p_2
  78. Segment(K, M) → q_2
  79. Segment(M, J) → r_2
  80. Segment(J, K) → s_2
  81. Segment(D_1, J_1) → t_2
  82. Segment(A_1, I_1) → a_3
  83. Segment(G_1, I_1) → b_3
  84. Segment(H_1, J_1) → c_3
  85. Segment(M, K) → d_3
  86. Segment(M, J) → e_3
  87. OrthogonalLine(I_1, d_3) → f_3
  88. Segment(O, N) → h_3
  89. Segment(M, O) → i_3
  90. OrthogonalLine(L_1, p_2) → k_3
  91. Intersect(k_3, i_3) → E_2
  92. Intersect(f_3, h_3) → F_2
  93. Segment(D_2, K_1) → g_3
  94. Segment(E_2, L_1) → j_3
  95. Segment(J_1, B_2) → l_3
  96. Segment(I_1, F_2) → m_3
  97. Segment(S, P) → n_3
  98. Segment(P, R) → p_3
  99. Segment(P, U) → q_3
  100. Segment(T, Q) → r_3
  101. Circle(J, 3) → s_3
  102. Circle(J, 4) → t_3
  103. Intersect(s_3, p) → G_2,
  104. Intersect(t_3, p) → H_2,
  105. Intersect(s_3, r_2) → I_2,
  106. Intersect(t_3, r_2) → J_2,
  107. OrthogonalLine(J_2, r_2) → a_4
  108. OrthogonalLine(I_2, r_2) → b_4
  109. OrthogonalLine(I_2, r_2) → c_4
  110. OrthogonalLine(G_1, r_2) → d_4
  111. OrthogonalLine(H_1, r_2) → e_4
  112. Intersect(a_4, q_2) → K_2
  113. Intersect(a_4, p_2) → L_2
  114. Intersect(b_4, c_4) → M_2
  115. Intersect(q_2, b_4) → N_2
  116. Intersect(p_2, b_4) → O_2
  117. OrthogonalLine(N_2, n_2) → f_4
  118. OrthogonalLine(K_2, q_2) → g_4
  119. OrthogonalLine(L_2, p_2) → h_4
  120. OrthogonalLine(O_2, p_2) → i_4
  121. Intersect(m_2, h_4) → P_2
  122. Intersect(i_4, m_2) → Q_2
  123. Intersect(n_2, f_4) → R_2
  124. Intersect(g_4, n_2) → S_2
  125. Segment(D_2, K_1) → j_4
  126. Segment(E_2, L_1) → k_4
  127. Segment(Q_2, O_2) → l_4
  128. Segment(P_2, L_2) → m_4
  129. Segment(F_2, I_1) → n_4
  130. Segment(B_2, J_1) → p_4
  131. Segment(J_1, H_2) → q_4
  132. Segment(I_1, G_2) → r_4
  133. Segment(N_2, D_1) → s_4
  134. Segment(K_2, A_1) → t_4
  135. Segment(H_1, J_1) → a_5
  136. Segment(G_1, I_1) → b_5
  137. Segment(I_2, N_2) → c_5
  138. Segment(J_2, K_2) → d_5
  139. Segment(K_2, S_2) → e_5
  140. Segment(R_2, N_2) → f_5
  141. Point(q_2) → T_2
  142. Point(q_2) → U_2
  143. Point(q_2) → V_2
  144. Point(q_2) → W_2
  145. Point(q_2) → Z_2
  146. Point(r_2) → A_3
  147. Point(p) → B_3
  148. Point(q_2) → C_3
  149. Point(q_2) → D_3
  150. Point(q_2) → E_3
  151. Point(q_2) → F_3
  152. Point(r_2) → G_3
  153. Point(p) → H_3
  154. Point(q_2) → I_3
  155. Point(q_2) → J_3
  156. Point(q_2) → K_3
  157. Point(r_2) → L_3
  158. Point(p) → M_3
  159. Point(q_2) → N_3
  160. Point(q_2) → O_3
  161. Point(r_2) → P_3
  162. Point(p) → Q_3
  163. Point(q_2) → R_3
  164. Point(r_2) → S_3
  165. Point(p) → T_3