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Visual GCD: Geometric Algorithm Demonstration

Visual GCD: Geometric Algorithm Demonstration
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2011-10-21 15:58
FunctionCoordinate GeometryGeometric Construction
This example demonstrates the calculation of the greatest common divisor using geometric figures. The core mathematical concepts involve number theory and geometric construction. The construction uses GeoGebra commands like If and Sequence to create objects. It is suitable for middle school students learning the combination of number theory and geometry.

GeoGebra Commands & Parameters

  1. If(A1 > B1, A1 - B1, A1) → A2
  2. If(A1 > B1, B1, B1 - A1) → B2
  3. If(A2 > B2, A2 - B2, A2) → A3
  4. If(A2 > B2, B2, B2 - A2) → B3
  5. If(A3 > B3, A3 - B3, A3) → A4
  6. If(A3 > B3, B3, B3 - A3) → B4
  7. If(A4 > B4, A4 - B4, A4) → A5
  8. If(A4 > B4, B4, B4 - A4) → B5
  9. If(A5 > B5, A5 - B5, A5) → A6
  10. If(A5 > B5, B5, B5 - A5) → B6
  11. If(A6 > B6, A6 - B6, A6) → A7
  12. If(A6 > B6, B6, B6 - A6) → B7
  13. If(A7 > B7, A7 - B7, A7) → A8
  14. If(A7 > B7, B7, B7 - A7) → B8
  15. If(A8 > B8, A8 - B8, A8) → A9
  16. If(A8 > B8, B8, B8 - A8) → B9
  17. If(A9 > B9, A9 - B9, A9) → A10
  18. If(A9 > B9, B9, B9 - A9) → B10
  19. If(A10 > B10, A10 - B10, A10) → A11
  20. If(A10 > B10, B10, B10 - A10) → B11
  21. If(A11 > B11, A11 - B11, A11) → A12
  22. If(A11 > B11, B11, B11 - A11) → B12
  23. If(A12 > B12, A12 - B12, A12) → A13
  24. If(A12 > B12, B12, B12 - A12) → B13
  25. If(A13 > B13, A13 - B13, A13) → A14
  26. If(A13 > B13, B13, B13 - A13) → B14
  27. If(A14 > B14, A14 - B14, A14) → A15
  28. If(A14 > B14, B14, B14 - A14) → B15
  29. If(A15 > B15, A15 - B15, A15) → A16
  30. If(A15 > B15, B15, B15 - A15) → B16
  31. If(A16 > B16, A16 - B16, A16) → A17
  32. If(A16 > B16, B16, B16 - A16) → B17
  33. If(A17 > B17, A17 - B17, A17) → A18
  34. If(A17 > B17, B17, B17 - A17) → B18
  35. If(A18 > B18, A18 - B18, A18) → A19
  36. If(A18 > B18, B18, B18 - A18) → B19
  37. If(A19 > B19, A19 - B19, A19) → A20
  38. If(A19 > B19, B19, B19 - A19) → B20
  39. If(A20 > B20, A20 - B20, A20) → A21
  40. If(A20 > B20, B20, B20 - A20) → B21
  41. If(A21 > B21, A21 - B21, A21) → A22
  42. If(A21 > B21, B21, B21 - A21) → B22
  43. If(A22 > B22, A22 - B22, A22) → A23
  44. If(A22 > B22, B22, B22 - A22) → B23
  45. If(A23 > B23, A23 - B23, A23) → A24
  46. If(A23 > B23, B23, B23 - A23) → B24
  47. If(A24 > B24, A24 - B24, A24) → A25
  48. If(A24 > B24, B24, B24 - A24) → B25
  49. If(A25 > B25, A25 - B25, A25) → A26
  50. If(A25 > B25, B25, B25 - A25) → B26
  51. If(A26 > B26, A26 - B26, A26) → A27
  52. If(A26 > B26, B26, B26 - A26) → B27
  53. If(A27 > B27, A27 - B27, A27) → A28
  54. If(A27 > B27, B27, B27 - A27) → B28
  55. If(A28 > B28, A28 - B28, A28) → A29
  56. If(A28 > B28, B28, B28 - A28) → B29
  57. If(A29 > B29, A29 - B29, A29) → A30
  58. If(A29 > B29, B29, B29 - A29) → B30
  59. If(A30 > B30, A30 - B30, A30) → A31
  60. If(A30 > B30, B30, B30 - A30) → B31
  61. If(A31 > B31, A31 - B31, A31) → A32
  62. If(A31 > B31, B31, B31 - A31) → B32
  63. If(A32 > B32, A32 - B32, A32) → A33
  64. If(A32 > B32, B32, B32 - A32) → B33
  65. If(A33 > B33, A33 - B33, A33) → A34
  66. If(A33 > B33, B33, B33 - A33) → B34
  67. If(A34 > B34, A34 - B34, A34) → A35
  68. If(A34 > B34, B34, B34 - A34) → B35
  69. If(A35 > B35, A35 - B35, A35) → A36
  70. If(A35 > B35, B35, B35 - A35) → B36
  71. If(A36 > B36, A36 - B36, A36) → A37
  72. If(A36 > B36, B36, B36 - A36) → B37
  73. If(A37 > B37, A37 - B37, A37) → A38
  74. If(A37 > B37, B37, B37 - A37) → B38
  75. If(A38 > B38, A38 - B38, A38) → A39
  76. If(A38 > B38, B38, B38 - A38) → B39
  77. If(A39 > B39, A39 - B39, A39) → A40
  78. If(A39 > B39, B39, B39 - A39) → B40
  79. If(A40 > B40, A40 - B40, A40) → A41
  80. If(A40 > B40, B40, B40 - A40) → B41
  81. If(A41 > B41, A41 - B41, A41) → A42
  82. If(A41 > B41, B41, B41 - A41) → B42
  83. If(A42 > B42, A42 - B42, A42) → A43
  84. If(A42 > B42, B42, B42 - A42) → B43
  85. If(A43 > B43, A43 - B43, A43) → A44
  86. If(A43 > B43, B43, B43 - A43) → B44
  87. If(A44 > B44, A44 - B44, A44) → A45
  88. If(A44 > B44, B44, B44 - A44) → B45
  89. If(A45 > B45, A45 - B45, A45) → A46
  90. If(A45 > B45, B45, B45 - A45) → B46
  91. If(A46 > B46, A46 - B46, A46) → A47
  92. If(A46 > B46, B46, B46 - A46) → B47
  93. If(A47 > B47, A47 - B47, A47) → A48
  94. If(A47 > B47, B47, B47 - A47) → B48
  95. If(A48 > B48, A48 - B48, A48) → A49
  96. If(A48 > B48, B48, B48 - A48) → B49
  97. If(A49 > B49, A49 - B49, A49) → A50
  98. If(A49 > B49, B49, B49 - A49) → B50
  99. If(A50 > B50, A50 - B50, A50) → A51
  100. If(A50 > B50, B50, B50 - A50) → B51
  101. If(A51 > B51, A51 - B51, A51) → A52
  102. If(A51 > B51, B51, B51 - A51) → B52
  103. If(A52 > B52, A52 - B52, A52) → A53
  104. If(A52 > B52, B52, B52 - A52) → B53
  105. If(A53 > B53, A53 - B53, A53) → A54
  106. If(A53 > B53, B53, B53 - A53) → B54
  107. If(A54 > B54, A54 - B54, A54) → A55
  108. If(A54 > B54, B54, B54 - A54) → B55
  109. If(A55 > B55, A55 - B55, A55) → A56
  110. If(A55 > B55, B55, B55 - A55) → B56
  111. If(A56 > B56, A56 - B56, A56) → A57
  112. If(A56 > B56, B56, B56 - A56) → B57
  113. If(A57 > B57, A57 - B57, A57) → A58
  114. If(A57 > B57, B57, B57 - A57) → B58
  115. If(A58 > B58, A58 - B58, A58) → A59
  116. If(A58 > B58, B58, B58 - A58) → B59
  117. If(A59 > B59, A59 - B59, A59) → A60
  118. If(A59 > B59, B59, B59 - A59) → B60
  119. If(A60 > B60, A60 - B60, A60) → A61
  120. If(A60 > B60, B60, B60 - A60) → B61
  121. If(A61 > B61, A61 - B61, A61) → A62
  122. If(A61 > B61, B61, B61 - A61) → B62
  123. If(A62 > B62, A62 - B62, A62) → A63
  124. If(A62 > B62, B62, B62 - A62) → B63
  125. If(A63 > B63, A63 - B63, A63) → A64
  126. If(A63 > B63, B63, B63 - A63) → B64
  127. If(A64 > B64, A64 - B64, A64) → A65
  128. If(A64 > B64, B64, B64 - A64) → B65
  129. If(A65 > B65, A65 - B65, A65) → A66
  130. If(A65 > B65, B65, B65 - A65) → B66
  131. If(A66 > B66, A66 - B66, A66) → A67
  132. If(A66 > B66, B66, B66 - A66) → B67
  133. If(A67 > B67, A67 - B67, A67) → A68
  134. If(A67 > B67, B67, B67 - A67) → B68
  135. If(A68 > B68, A68 - B68, A68) → A69
  136. If(A68 > B68, B68, B68 - A68) → B69
  137. If(A69 > B69, A69 - B69, A69) → A70
  138. If(A69 > B69, B69, B69 - A69) → B70
  139. If(A70 > B70, A70 - B70, A70) → A71
  140. If(A70 > B70, B70, B70 - A70) → B71
  141. If(A71 > B71, A71 - B71, A71) → A72
  142. If(A71 > B71, B71, B71 - A71) → B72
  143. If(A72 > B72, A72 - B72, A72) → A73
  144. If(A72 > B72, B72, B72 - A72) → B73
  145. If(A73 > B73, A73 - B73, A73) → A74
  146. If(A73 > B73, B73, B73 - A73) → B74
  147. If(A74 > B74, A74 - B74, A74) → A75
  148. If(A74 > B74, B74, B74 - A74) → B75
  149. If(A75 > B75, A75 - B75, A75) → A76
  150. If(A75 > B75, B75, B75 - A75) → B76
  151. If(A76 > B76, A76 - B76, A76) → A77
  152. If(A76 > B76, B76, B76 - A76) → B77
  153. If(A77 > B77, A77 - B77, A77) → A78
  154. If(A77 > B77, B77, B77 - A77) → B78
  155. Sequence(Segment[(Cell[1, i], Cell[2, i]), (Cell[1, i], 0)], i, 1, 100) → list1
  156. Sequence(Segment[(Cell[1, i], Cell[2, i]), (0, Cell[2, i])], i, 1, 100) → list2
  157. If(A78 > B78, A78 - B78, A78) → A79
  158. If(A78 > B78, B78, B78 - A78) → B79
  159. If(A79 > B79, A79 - B79, A79) → A80
  160. If(A79 > B79, B79, B79 - A79) → B80
  161. If(A80 > B80, A80 - B80, A80) → A81
  162. If(A80 > B80, B80, B80 - A80) → B81
  163. If(A81 > B81, A81 - B81, A81) → A82
  164. If(A81 > B81, B81, B81 - A81) → B82
  165. If(A82 > B82, A82 - B82, A82) → A83
  166. If(A82 > B82, B82, B82 - A82) → B83
  167. If(A83 > B83, A83 - B83, A83) → A84
  168. If(A83 > B83, B83, B83 - A83) → B84
  169. If(A84 > B84, A84 - B84, A84) → A85
  170. If(A84 > B84, B84, B84 - A84) → B85
  171. If(A85 > B85, A85 - B85, A85) → A86
  172. If(A85 > B85, B85, B85 - A85) → B86
  173. If(A86 > B86, A86 - B86, A86) → A87
  174. If(A86 > B86, B86, B86 - A86) → B87
  175. If(A87 > B87, A87 - B87, A87) → A88
  176. If(A87 > B87, B87, B87 - A87) → B88
  177. If(A88 > B88, A88 - B88, A88) → A89
  178. If(A88 > B88, B88, B88 - A88) → B89
  179. If(A89 > B89, A89 - B89, A89) → A90
  180. If(A89 > B89, B89, B89 - A89) → B90
  181. If(A90 > B90, A90 - B90, A90) → A91
  182. If(A90 > B90, B90, B90 - A90) → B91
  183. If(A91 > B91, A91 - B91, A91) → A92
  184. If(A91 > B91, B91, B91 - A91) → B92
  185. If(A92 > B92, A92 - B92, A92) → A93
  186. If(A92 > B92, B92, B92 - A92) → B93
  187. If(A93 > B93, A93 - B93, A93) → A94
  188. If(A93 > B93, B93, B93 - A93) → B94
  189. If(A94 > B94, A94 - B94, A94) → A95
  190. If(A94 > B94, B94, B94 - A94) → B95
  191. If(A95 > B95, A95 - B95, A95) → A96
  192. If(A95 > B95, B95, B95 - A95) → B96
  193. If(A96 > B96, A96 - B96, A96) → A97
  194. If(A96 > B96, B96, B96 - A96) → B97
  195. If(A97 > B97, A97 - B97, A97) → A98
  196. If(A97 > B97, B97, B97 - A97) → B98
  197. If(A98 > B98, A98 - B98, A98) → A99
  198. If(A98 > B98, B98, B98 - A98) → B99
  199. If(A99 > B99, A99 - B99, A99) → A100
  200. If(A99 > B99, B99, B99 - A99) → B100
  201. Sequence(Ray[(0, 5 * i), (1, 5 * i)], i, 1, 20) → list3
  202. Sequence(Ray[(5 * i, 0), (5 * i, 1)], i, 1, 20) → list4

AI Drawing Prompt Reference

Draw a geometric algorithm diagram for finding the greatest common divisor (GCD) of two positive integers a and b. Start by setting a and b as two points, then repeatedly perform the operation of subtracting the smaller number from the larger one until the result is less than or equal to the other number. Continue this process until the result is 0, which is the GCD. In the diagram, you will see a series of subtraction operations forming a decreasing sequence until the GCD is found.
The prompt above was compiled from the original drawing commands and is for reference only.