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Proof without words: The third power sum

Proof without words: The third power sum
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2019-12-28 15:54
SequenceCoordinate GeometryGeometric Construction
This example demonstrates the proof of the sum of cubes through geometric figures. It involves concepts such as cubes, parabolas, geometric means, and symmetry. Using GeoGebra commands like Midpoint and Expand, it constructs multiple points and objects to illustrate the geometric meaning of the sum of cubes. Suitable for high school and above students to learn the geometric proof of the sum of cubes.

GeoGebra Commands & Parameters

  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