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无言证明：立方和

无言证明：立方和
几何画板及绘图内容加载中，请稍等……
2019-12-28 15:54
数列解析几何几何作图
这个范例通过几何图形展示了立方和的证明。它涉及立方、抛物线、几何均值、对称性等核心数学概念。通过使用Midpoint、Expand等GeoGebra命令，构造了多个点和对象，展示了立方和的几何意义。适合高中及以上学生学习立方和的几何证明。

GeoGebra 命令及参数

  1. If(Angle[F1, G1, H1] < 180°, Polygon[F1, G1, H1, I1]) → J1
  2. If(Angle[F1, G1, H1] < 180°, Polygon[F1, G1, H1, I1]) → K1
  3. If(Angle[F2, G2, H2] < 180°, Polygon[F2, G2, H2, I2]) → J2
  4. If(Angle[F2, G2, H2] < 180°, Polygon[F2, G2, H2, I2]) → K2
  5. If(Angle[F3, G3, H3] < 180°, Polygon[F3, G3, H3, I3]) → J3
  6. If(Angle[F3, G3, H3] < 180°, Polygon[F3, G3, H3, I3]) → K3
  7. If(Angle[L1, M1, N1] < 180°, Polygon[L1, M1, N1, O1]) → P1
  8. If(Angle[L1, M1, N1] < 180°, Polygon[L1, M1, N1, O1]) → Q1
  9. If(Angle[L2, M2, N2] < 180°, Polygon[L2, M2, N2, O2]) → P2
  10. If(Angle[L2, M2, N2] < 180°, Polygon[L2, M2, N2, O2]) → Q2
  11. If(Angle[L3, M3, N3] < 180°, Polygon[L3, M3, N3, O3]) → P3
  12. If(Angle[L3, M3, N3] < 180°, Polygon[L3, M3, N3, O3]) → Q3
  13. If(Angle[F4, G4, H4] < 180°, Polygon[F4, G4, H4, I4]) → J4
  14. If(Angle[F4, G4, H4] < 180°, Polygon[F4, G4, H4, I4]) → K4
  15. If(Angle[F5, G5, H5] < 180°, Polygon[F5, G5, H5, I5]) → J5
  16. If(Angle[F5, G5, H5] < 180°, Polygon[F5, G5, H5, I5]) → K5
  17. If(Angle[F6, G6, H6] < 180°, Polygon[F6, G6, H6, I6]) → J6
  18. If(Angle[F6, G6, H6] < 180°, Polygon[F6, G6, H6, I6]) → K6
  19. If(Angle[L4, M4, N4] < 180°, Polygon[L4, M4, N4, O4]) → P4
  20. If(Angle[L4, M4, N4] < 180°, Polygon[L4, M4, N4, O4]) → Q4
  21. If(Angle[L5, M5, N5] < 180°, Polygon[L5, M5, N5, O5]) → P5
  22. If(Angle[L5, M5, N5] < 180°, Polygon[L5, M5, N5, O5]) → Q5
  23. If(Angle[L6, M6, N6] < 180°, Polygon[L6, M6, N6, O6]) → P6
  24. If(Angle[L6, M6, N6] < 180°, Polygon[L6, M6, N6, O6]) → Q6
  25. If(Angle[F7, G7, H7] < 180°, Polygon[F7, G7, H7, I7]) → J7
  26. If(Angle[F7, G7, H7] < 180°, Polygon[F7, G7, H7, I7]) → K7
  27. If(Angle[F8, G8, H8] < 180°, Polygon[F8, G8, H8, I8]) → J8
  28. If(Angle[F8, G8, H8] < 180°, Polygon[F8, G8, H8, I8]) → K8
  29. If(Angle[F9, G9, H9] < 180°, Polygon[F9, G9, H9, I9]) → J9
  30. If(Angle[F9, G9, H9] < 180°, Polygon[F9, G9, H9, I9]) → K9
  31. If(Angle[L7, M7, N7] < 180°, Polygon[L7, M7, N7, O7]) → P7
  32. If(Angle[L7, M7, N7] < 180°, Polygon[L7, M7, N7, O7]) → Q7
  33. If(Angle[L8, M8, N8] < 180°, Polygon[L8, M8, N8, O8]) → P8
  34. If(Angle[L8, M8, N8] < 180°, Polygon[L8, M8, N8, O8]) → Q8
  35. If(Angle[L9, M9, N9] < 180°, Polygon[L9, M9, N9, O9]) → P9
  36. If(Angle[L9, M9, N9] < 180°, Polygon[L9, M9, N9, O9]) → Q9
  37. If(Angle[F10, G10, H10] < 180°, Polygon[F10, G10, H10, I10]) → J10
  38. If(Angle[F10, G10, H10] < 180°, Polygon[F10, G10, H10, I10]) → K10
  39. If(Angle[F11, G11, H11] < 180°, Polygon[F11, G11, H11, I11]) → J11
  40. If(Angle[F11, G11, H11] < 180°, Polygon[F11, G11, H11, I11]) → K11
  41. If(Angle[F12, G12, H12] < 180°, Polygon[F12, G12, H12, I12]) → J12
  42. If(Angle[F12, G12, H12] < 180°, Polygon[F12, G12, H12, I12]) → K12
  43. If(Angle[L10, M10, N10] < 180°, Polygon[L10, M10, N10, O10]) → P10
  44. If(Angle[L10, M10, N10] < 180°, Polygon[L10, M10, N10, O10]) → Q10
  45. If(Angle[L11, M11, N11] < 180°, Polygon[L11, M11, N11, O11]) → P11
  46. If(Angle[L11, M11, N11] < 180°, Polygon[L11, M11, N11, O11]) → Q11
  47. If(Angle[L12, M12, N12] < 180°, Polygon[L12, M12, N12, O12]) → P12
  48. If(Angle[L12, M12, N12] < 180°, Polygon[L12, M12, N12, O12]) → Q12
  49. If(Angle[F13, G13, H13] < 180°, Polygon[F13, G13, H13, I13]) → J13
  50. If(Angle[F13, G13, H13] < 180°, Polygon[F13, G13, H13, I13]) → K13
  51. If(Angle[F14, G14, H14] < 180°, Polygon[F14, G14, H14, I14]) → J14
  52. If(Angle[F14, G14, H14] < 180°, Polygon[F14, G14, H14, I14]) → K14
  53. If(Angle[F15, G15, H15] < 180°, Polygon[F15, G15, H15, I15]) → J15
  54. If(Angle[F15, G15, H15] < 180°, Polygon[F15, G15, H15, I15]) → K15
  55. If(Angle[L13, M13, N13] < 180°, Polygon[L13, M13, N13, O13]) → P13
  56. If(Angle[L13, M13, N13] < 180°, Polygon[L13, M13, N13, O13]) → Q13
  57. If(Angle[L14, M14, N14] < 180°, Polygon[L14, M14, N14, O14]) → P14
  58. If(Angle[L14, M14, N14] < 180°, Polygon[L14, M14, N14, O14]) → Q14
  59. If(Angle[L15, M15, N15] < 180°, Polygon[L15, M15, N15, O15]) → P15
  60. If(Angle[L15, M15, N15] < 180°, Polygon[L15, M15, N15, O15]) → Q15
  61. If(Angle[F16, G16, H16] < 180°, Polygon[F16, G16, H16, I16]) → J16
  62. If(Angle[F16, G16, H16] < 180°, Polygon[F16, G16, H16, I16]) → K16
  63. If(Angle[F17, G17, H17] < 180°, Polygon[F17, G17, H17, I17]) → J17
  64. If(Angle[F17, G17, H17] < 180°, Polygon[F17, G17, H17, I17]) → K17
  65. If(Angle[F18, G18, H18] < 180°, Polygon[F18, G18, H18, I18]) → J18
  66. If(Angle[F18, G18, H18] < 180°, Polygon[F18, G18, H18, I18]) → K18
  67. If(Angle[L16, M16, N16] < 180°, Polygon[L16, M16, N16, O16]) → P16
  68. If(Angle[L16, M16, N16] < 180°, Polygon[L16, M16, N16, O16]) → Q16
  69. If(Angle[L17, M17, N17] < 180°, Polygon[L17, M17, N17, O17]) → P17
  70. If(Angle[L17, M17, N17] < 180°, Polygon[L17, M17, N17, O17]) → Q17
  71. If(Angle[L18, M18, N18] < 180°, Polygon[L18, M18, N18, O18]) → P18
  72. If(Angle[L18, M18, N18] < 180°, Polygon[L18, M18, N18, O18]) → Q18
  73. If(Angle[F19, G19, H19] < 180°, Polygon[F19, G19, H19, I19]) → J19
  74. If(Angle[F19, G19, H19] < 180°, Polygon[F19, G19, H19, I19]) → K19
  75. If(Angle[F20, G20, H20] < 180°, Polygon[F20, G20, H20, I20]) → J20
  76. If(Angle[F20, G20, H20] < 180°, Polygon[F20, G20, H20, I20]) → K20
  77. If(Angle[F21, G21, H21] < 180°, Polygon[F21, G21, H21, I21]) → J21
  78. If(Angle[F21, G21, H21] < 180°, Polygon[F21, G21, H21, I21]) → K21
  79. If(Angle[L19, M19, N19] < 180°, Polygon[L19, M19, N19, O19]) → P19
  80. If(Angle[L19, M19, N19] < 180°, Polygon[L19, M19, N19, O19]) → Q19
  81. If(Angle[L20, M20, N20] < 180°, Polygon[L20, M20, N20, O20]) → P20
  82. If(Angle[L20, M20, N20] < 180°, Polygon[L20, M20, N20, O20]) → Q20
  83. If(Angle[L21, M21, N21] < 180°, Polygon[L21, M21, N21, O21]) → P21
  84. If(Angle[L21, M21, N21] < 180°, Polygon[L21, M21, N21, O21]) → Q21
  85. If(Angle[F22, G22, H22] < 180°, Polygon[F22, G22, H22, I22]) → J22
  86. If(Angle[F22, G22, H22] < 180°, Polygon[F22, G22, H22, I22]) → K22
  87. If(Angle[F23, G23, H23] < 180°, Polygon[F23, G23, H23, I23]) → J23
  88. If(Angle[F23, G23, H23] < 180°, Polygon[F23, G23, H23, I23]) → K23
  89. If(Angle[F24, G24, H24] < 180°, Polygon[F24, G24, H24, I24]) → J24
  90. If(Angle[F24, G24, H24] < 180°, Polygon[F24, G24, H24, I24]) → K24
  91. If(Angle[L22, M22, N22] < 180°, Polygon[L22, M22, N22, O22]) → P22
  92. If(Angle[L22, M22, N22] < 180°, Polygon[L22, M22, N22, O22]) → Q22
  93. If(Angle[L23, M23, N23] < 180°, Polygon[L23, M23, N23, O23]) → P23
  94. If(Angle[L23, M23, N23] < 180°, Polygon[L23, M23, N23, O23]) → Q23
  95. If(Angle[L24, M24, N24] < 180°, Polygon[L24, M24, N24, O24]) → P24
  96. If(Angle[L24, M24, N24] < 180°, Polygon[L24, M24, N24, O24]) → Q24
  97. Defined(P1) → R1
  98. Defined(P2) → R2
  99. Defined(P3) → R3
  100. Defined(P4) → R4
  101. Defined(P5) → R5
  102. Defined(P6) → R6
  103. Defined(P7) → R7
  104. Defined(P8) → R8
  105. Defined(P9) → R9
  106. Defined(P10) → R10
  107. Defined(P11) → R11
  108. Defined(P12) → R12
  109. Defined(P13) → R13
  110. Defined(P14) → R14
  111. Defined(P15) → R15
  112. Defined(P16) → R16
  113. Defined(P17) → R17
  114. Defined(P18) → R18
  115. Defined(P19) → R19
  116. Defined(P20) → R20
  117. Defined(P21) → R21
  118. Defined(P22) → R22
  119. Defined(P23) → R23
  120. Defined(P24) → R24
  121. If(S1 ∨ T1, P1) → U1
  122. If(S2 ∨ T2, P2) → U2
  123. If(S3 ∨ T3, P3) → U3
  124. If(S4 ∨ T4, P4) → U4
  125. If(S5 ∨ T5, P5) → U5
  126. If(S6 ∨ T6, P6) → U6
  127. If(S7 ∨ T7, P7) → U7
  128. If(S8 ∨ T8, P8) → U8
  129. If(S9 ∨ T9, P9) → U9
  130. If(S10 ∨ T10, P10) → U10
  131. If(S11 ∨ T11, P11) → U11
  132. If(S12 ∨ T12, P12) → U12
  133. If(S13 ∨ T13, P13) → U13
  134. If(S14 ∨ T14, P14) → U14
  135. If(S15 ∨ T15, P15) → U15
  136. If(S16 ∨ T16, P16) → U16
  137. If(S17 ∨ T17, P17) → U17
  138. If(S18 ∨ T18, P18) → U18
  139. If(S19 ∨ T19, P19) → U19
  140. If(S20 ∨ T20, P20) → U20
  141. If(S21 ∨ T21, P21) → U21
  142. If(S22 ∨ T22, P22) → U22
  143. If(S23 ∨ T23, P23) → U23
  144. If(S24 ∨ T24, P24) → U24
  145. If(S1 ∨ T1, Q1) → V1
  146. If(S2 ∨ T2, Q2) → V2
  147. If(S3 ∨ T3, Q3) → V3
  148. If(S4 ∨ T4, Q4) → V4
  149. If(S5 ∨ T5, Q5) → V5
  150. If(S6 ∨ T6, Q6) → V6
  151. If(S7 ∨ T7, Q7) → V7
  152. If(S8 ∨ T8, Q8) → V8
  153. If(S9 ∨ T9, Q9) → V9
  154. If(S10 ∨ T10, Q10) → V10
  155. If(S11 ∨ T11, Q11) → V11
  156. If(S12 ∨ T12, Q12) → V12
  157. If(S13 ∨ T13, Q13) → V13
  158. If(S14 ∨ T14, Q14) → V14
  159. If(S15 ∨ T15, Q15) → V15
  160. If(S16 ∨ T16, Q16) → V16
  161. If(S17 ∨ T17, Q17) → V17
  162. If(S18 ∨ T18, Q18) → V18
  163. If(S19 ∨ T19, Q19) → V19
  164. If(S20 ∨ T20, Q20) → V20
  165. If(S21 ∨ T21, Q21) → V21
  166. If(S22 ∨ T22, Q22) → V22
  167. If(S23 ∨ T23, Q23) → V23
  168. If(S24 ∨ T24, Q24) → V24
  169. Defined(U1) → W1
  170. Defined(U2) → W2
  171. Defined(U3) → W3
  172. Defined(U4) → W4
  173. Defined(U5) → W5
  174. Defined(U6) → W6
  175. Defined(U7) → W7
  176. Defined(U8) → W8
  177. Defined(U9) → W9
  178. Defined(U10) → W10
  179. Defined(U11) → W11
  180. Defined(U12) → W12
  181. Defined(U13) → W13
  182. Defined(U14) → W14
  183. Defined(U15) → W15
  184. Defined(U16) → W16
  185. Defined(U17) → W17
  186. Defined(U18) → W18
  187. Defined(U19) → W19
  188. Defined(U20) → W20
  189. Defined(U21) → W21
  190. Defined(U22) → W22
  191. Defined(U23) → W23
  192. Defined(U24) → W24
  193. If(X1 ∨ Y1, U1) → Z1
  194. If(X2 ∨ Y2, U2) → Z2
  195. If(X3 ∨ Y3, U3) → Z3
  196. If(X4 ∨ Y4, U4) → Z4
  197. If(X5 ∨ Y5, U5) → Z5
  198. If(X6 ∨ Y6, U6) → Z6
  199. If(X7 ∨ Y7, U7) → Z7
  200. If(X8 ∨ Y8, U8) → Z8
  201. If(X9 ∨ Y9, U9) → Z9
  202. If(X10 ∨ Y10, U10) → Z10
  203. If(X11 ∨ Y11, U11) → Z11
  204. If(X12 ∨ Y12, U12) → Z12
  205. If(X13 ∨ Y13, U13) → Z13
  206. If(X14 ∨ Y14, U14) → Z14
  207. If(X15 ∨ Y15, U15) → Z15
  208. If(X16 ∨ Y16, U16) → Z16
  209. If(X17 ∨ Y17, U17) → Z17
  210. If(X18 ∨ Y18, U18) → Z18
  211. If(X19 ∨ Y19, U19) → Z19
  212. If(X20 ∨ Y20, U20) → Z20
  213. If(X21 ∨ Y21, U21) → Z21
  214. If(X22 ∨ Y22, U22) → Z22
  215. If(X23 ∨ Y23, U23) → Z23
  216. If(X24 ∨ Y24, U24) → Z24
  217. If(X1 ∨ Y1, V1) → AA1
  218. If(X2 ∨ Y2, V2) → AA2
  219. If(X3 ∨ Y3, V3) → AA3
  220. If(X4 ∨ Y4, V4) → AA4
  221. If(X5 ∨ Y5, V5) → AA5
  222. If(X6 ∨ Y6, V6) → AA6
  223. If(X7 ∨ Y7, V7) → AA7
  224. If(X8 ∨ Y8, V8) → AA8
  225. If(X9 ∨ Y9, V9) → AA9
  226. If(X10 ∨ Y10, V10) → AA10
  227. If(X11 ∨ Y11, V11) → AA11
  228. If(X12 ∨ Y12, V12) → AA12
  229. If(X13 ∨ Y13, V13) → AA13
  230. If(X14 ∨ Y14, V14) → AA14
  231. If(X15 ∨ Y15, V15) → AA15
  232. If(X16 ∨ Y16, V16) → AA16
  233. If(X17 ∨ Y17, V17) → AA17
  234. If(X18 ∨ Y18, V18) → AA18
  235. If(X19 ∨ Y19, V19) → AA19
  236. If(X20 ∨ Y20, V20) → AA20
  237. If(X21 ∨ Y21, V21) → AA21
  238. If(X22 ∨ Y22, V22) → AA22
  239. If(X24 ∨ Y24, V24) → AA24
  240. Midpoint(G6, F4) → A
  241. Midpoint(F7, G7) → B
  242. Midpoint(F10, G10) → D
  243. Midpoint(G9, F7) → E
  244. Midpoint(F1, G2) → F
  245. Midpoint(G4, F4) → G
  246. Midpoint(F4, G5) → H
  247. Midpoint(G18, F16) → I