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Geometric Construction of Ellipse and Circle in GeoGebra

Geometric Construction of Ellipse and Circle in GeoGebra
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2023-03-22 02:20
CircleGeometric Construction
In collections Realidades de GeoGebra
This example demonstrates constructing an ellipse and a circle using GeoGebra, analyzing their geometric properties. Key mathematical concepts include the definition of an ellipse, the equation of a circle, and geometric transformations. Construction steps involve using Ellipse, Circle, Point, and Translate commands, along with Midpoint and Vector for geometric analysis. Suitable for high school students or college freshmen learning coordinate geometry and geometric construction.

GeoGebra Commands & Parameters

  1. Ellipse(O_r - (0.95, 0), O_r + (0.95, 0), 1) → mira1
  2. Point(mira1) → A
  3. Ellipse(O_r - (0, 0.95), O_r + (0, 0.95), 1) → mira2
  4. Point(mira2) → B
  5. punto3d(A, B, M, O_0, O_r, -4, 0.12121212121212122, 1, U, L) → E
  6. punto3d(A, B, M, O_0, O_r, -4, 1, 1, U, L) → V5
  7. punto3d(A, B, M, O_0, O_r, 1, -1, 1, U, L) → V3
  8. punto3d(A, B, M, O_0, O_r, 1, 1, 1, U, L) → V2
  9. Segment(V2, V5) → cubA8
  10. Segment(V2, V3) → cubA2
  11. punto3d(A, B, M, O_0, O_r, -4, 3, -1, U, L) → P
  12. punto3d(A, B, M, O_0, O_r, 1, 3, -1, U, L) → W
  13. Line(W, P) → l1
  14. punto3d(A, B, M, O_0, O_r, -4.166666666666667, 4, -1, U, L) → C
  15. punto3d(A, B, M, O_0, O_r, 1, 0.12121212121212122, -1, U, L) → D
  16. Line(D, C) → f1
  17. Intersect(f1, l1) → S
  18. Line(S, cubA2) → i1
  19. punto3d(A, B, M, O_0, O_r, 1, 1, -1, U, L) → V1
  20. Segment(V1, V2) → cubA1
  21. punto3d(A, B, M, O_0, O_r, -4, 1, -1, U, L) → V6
  22. Segment(V1, V6) → cubA5
  23. Intersect(cubA5, i1) → T
  24. Line(T, cubA1) → j1
  25. Intersect(cubA8, j1) → V
  26. Intersect(f1, cubA5) → K
  27. punto3d(A, B, M, O_0, O_r, ((-sqrt(4 - (t - 2)^(2)))) / 12 - 4, (2 * t), (t - 14) / 12 + sqrt(4 - t^(2)) + sqrt((8 * t) - (4 * t^(2))) / 2, U, L) → J
  28. punto3d(A, B, M, O_0, O_r, 1 + t / 12, 0, sqrt(4 - t^(2)) / 12 - 1, U, L) → R1a'
  29. punto3d(A, B, M, O_0, O_r, -4 - (t * 11 / 12), 0, (sqrt(4 - t^(2)) * 11 / 12) - 1, U, L) → R1c'
  30. punto3d(A, B, M, O_0, O_r, -4, 0, -1, U, L) → R1b
  31. punto3d(A, B, M, O_0, O_r, 1, 0, -1, U, L) → R1a
  32. punto3d(A, B, M, O_0, O_r, 1, 0, -0.8333333333333334, U, L) → Araña
  33. Intersect(mira1, mira2) → Mira1, Mira2, Mira3, Mira4
  34. Angle(M, O_r, B) → θ
  35. Angle(U, O_r, A) → φ
  36. Vector(O_r, O_0) → u
  37. Translate(Xr, u) → X0
  38. Translate(Zr, u) → Z0
  39. Translate(Yr, u) → Y0
  40. Vector(O_0, Z0) → v
  41. Vector(O_0, Y0) → w
  42. Vector(O_0, X0) → z
  43. Circle(O_0, L) → c
  44. Line(O_0, Y0) → a
  45. Line(O_0, yAxis) → b
  46. Line(O_0, X0) → g
  47. Line(Z0, O_0) → j
  48. Translate(P1r, u) → P1
  49. Rotate(A, botonpaso, O_r) → botonA
  50. Rotate(B, botonpaso, O_r) → botonB
  51. Segment(Mira3, Mira2) → k
  52. Segment((-1, 0), U) → l
  53. Segment((0, -1), M) → m
  54. punto3d(A, B, M, O_0, O_r, 1, -1, -1, U, L) → V4
  55. punto3d(A, B, M, O_0, O_r, -4, -1, 1, U, L) → V7
  56. punto3d(A, B, M, O_0, O_r, -4, -1, -1, U, L) → V8
  57. Midpoint(V1, V2) → cub1M
  58. Midpoint(V3, V2) → cub2M
  59. Midpoint(V1, V6) → cub4M
  60. Midpoint(V7, V8) → cub5M
  61. Angle(V3, cub1M, V4) → cub1Ang
  62. If(cub1Ang < 180°, V2, V3) → cubV2b
  63. If(cub1Ang < 180°, V1, V4) → cubV1b
  64. Polygon(V3, V4, cubV1b, cubV2b) → cub1, cub1v3, cub1v4, cub1v1, cub1v2
  65. Angle(V6, cub1M, V5) → cub3Ang
  66. If(cub3Ang < 180°, V2, V5) → cubV2c
  67. If(cub3Ang < 180°, V1, V6) → cubV1c
  68. Polygon(V6, V5, cubV2c, cubV1c) → cub3, cub3v6, cub3v5, cub3v2, cub3v1
  69. Angle(V5, cub5M, V6) → cub5Ang
  70. If(cub5Ang < 180°, V5, V7) → cubV5b
  71. If(cub5Ang < 180°, V6, V8) → cubV6b
  72. Polygon(V8, V7, cubV5b, cubV6b) → cub5, cub5v8, cub5v7, cub5v5, cub5v6
  73. Angle(V4, cub5M, V3) → cub6Ang
  74. If(cub6Ang < 180°, V7, V3) → cubV7b
  75. If(cub6Ang < 180°, V8, V4) → cubV8b
  76. Polygon(V4, V3, cubV7b, cubV8b) → cub6, cub6v4, cub6v3, cub6v7, cub6v8
  77. Angle(V5, cub2M, V7) → cub2Ang
  78. If(cub2Ang < 180°, V3, V7) → cubV3b
  79. If(cub2Ang < 180°, V2, V5) → cubV2d
  80. Polygon(V7, V5, cubV2d, cubV3b) → cub2, cub2v7, cub2v5, cub2v2, cub2v3
  81. Angle(V4, cub4M, V8) → cub4Ang
  82. Segment(V2, V1) → cubA1'
  83. Segment(V2, V3) → cubA2'
  84. Segment(V3, V4) → cubA3'
  85. Segment(V4, V1) → cubA4'
  86. Segment(V1, V6) → cubA5'
  87. Segment(V8, V6) → cubA6'
  88. Segment(V4, V8) → cubA7'
  89. Segment(V2, V5) → cubA8'
  90. Segment(V5, V7) → cubA9'
  91. Segment(V3, V7) → cubA10'
  92. Segment(V5, V6) → cubA11'
  93. Segment(V7, V8) → cubA12'
  94. Segment(V3, V4) → cubA3
  95. Segment(V4, V1) → cubA4
  96. Segment(V6, V8) → cubA6
  97. Segment(V8, V4) → cubA7
  98. Segment(V5, V7) → cubA9
  99. Segment(V7, V3) → cubA10
  100. Segment(V6, V5) → cubA11
  101. Segment(V7, V8) → cubA12
  102. If(cub4Ang < 180°, V1, V4) → cubV1d
  103. If(cub4Ang < 180°, V6, V8) → cubV6c
  104. Polygon(V8, V4, cubV1d, cubV6c) → cub4, cub4v8, cub4v4, cub4v1, cub4v6
  105. punto3d(A, B, M, O_0, O_r, 1, 1 + t, sqrt(4 - t^(2)) - 1, U, L) → V2b
  106. punto3d(A, B, M, O_0, O_r, 1, -1 - t, sqrt(4 - t^(2)) - 1, U, L) → V3c
  107. punto3d(A, B, M, O_0, O_r, 1 + t, -1, sqrt(4 - t^(2)) - 1, U, L) → V3d
  108. punto3d(A, B, M, O_0, O_r, 1 + t, 1, sqrt(4 - t^(2)) - 1, U, L) → V2d
  109. Polygon(V4, V3d, V2d, V1) → poly2, v4, v3d_1, v2d_1, v1
  110. Polygon(V1, V4, V8, V6) → poly6, v1_2, v4_2, v8_2, v6_2
  111. punto3d(A, B, M, O_0, O_r, 1, (3 * t) - 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V3f
  112. punto3d(A, B, M, O_0, O_r, 1, (-3 * t) + 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V2e
  113. punto3d(A, B, M, O_0, O_r, -4, -1 - t, sqrt(4 - t^(2)) - 1, U, L) → V7c
  114. punto3d(A, B, M, O_0, O_r, -4, (-3 * t) + 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V5e
  115. punto3d(A, B, M, O_0, O_r, -4, 1 + t, sqrt(4 - t^(2)) - 1, U, L) → V5b
  116. punto3d(A, B, M, O_0, O_r, -4 - t, -1, sqrt(4 - t^(2)) - 1, U, L) → V7a
  117. punto3d(A, B, M, O_0, O_r, -4 - t, 1, sqrt(4 - t^(2)) - 1, U, L) → V5a
  118. punto3d(A, B, M, O_0, O_r, -4, (3 * t) - 1, sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 1, U, L) → V7f
  119. punto3d(A, B, M, O_0, O_r, (-sqrt(4 - (t - 2)^(2))) - 4, 1 + t, t + sqrt(4 - t^(2)) - 3, U, L) → v6g
  120. punto3d(A, B, M, O_0, O_r, (-sqrt(4 - (t - 2)^(2))) - 4, (3 * t) - 1, t + sqrt((8 * t) - (4 * t^(2))) + sqrt(4 - t^(2)) - 3, U, L) → V8g
  121. Polygon(V4, V3c, V7c, V8) → poly1, v4_1, v3c, v7c, v8
  122. Polygon(V3c, V2e, V5e, V7c) → poly3, v3c_1, v2e, v5e, v7c_1
  123. Polygon(V8, V7a, V5a, V6) → poly4, v8_1, v7a, v5a, v6
  124. Polygon(V6, V5b, V2b, V1) → poly5, v6_1, v5b, v2b, v1_1
  125. Polygon(V5b, V7f, V3f, V2b) → poly7, v5b_1, v7f, v3f, v2b_1
  126. Polygon(V5b, v6g, V8g, V7f) → poly8, v5b_2, v6g_1, v8g, v7f_1
  127. punto3d(A, B, M, O_0, O_r, -4, 0, 0.8333333333333334, U, L) → Mosca
  128. Segment(R1b, Mosca) → r1c
  129. Segment(Araña, R1a) → r1a
  130. Segment(R1a, R1b) → r1b
  131. Segment(R1a', R1a) → r1a'
  132. Segment(R1b, R1c') → r1c'
  133. Segment(V2, V6) → d
  134. Segment(V1, V5) → e
  135. Midpoint(d) → F
  136. Segment(V2b, V6) → f
  137. Segment(V1, V5b) → h
  138. Midpoint(f) → I
  139. Segment(R1a', D) → i
  140. Segment(Araña, D) → s
  141. Segment(R1a, R1b) → e1
  142. Segment(K, V) → k1
  143. Segment(E, Mosca) → d1
  144. Line(T, v6_1) → m1
  145. Intersect(v5b, v2b_1) → Z
  146. Intersect(v5b, v2b_1) → A1
  147. Intersect(v5b, v2b_1) → B1
  148. Intersect(m1, v5b) → C1
  149. Segment(D, K) → n
  150. Segment(K, C1) → q
  151. Segment(V, E) → b1
  152. punto3d(A, B, M, O_0, O_r, -4, 1 + t + (29 * (t - 1) / 33), sqrt(4 - t^(2)) - 1 + (29 * sqrt((2 * t) - t^(2)) / 33), U, L) → G
  153. Segment(G, J) → p
  154. Segment(C1, G) → r
  155. Segment(D, K) → a1

AI Drawing Prompt Reference

Draw an ellipse and a circle to demonstrate their geometric properties. Begin by constructing two perpendicular ellipses, then find their intersection points. Next, use these points and points on the ellipses to create lines and segments, forming a geometric pattern. Finally, add auxiliary points and lines on the ellipses and circle to highlight geometric transformations and coordinate geometry concepts.
The prompt above was compiled from the original drawing commands and is for reference only.