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Equal Areas in Ellipse

Equal Areas in Ellipse
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2019-03-19 05:24
Coordinate GeometryGeometric Construction
This example demonstrates the property of equal areas in an ellipse. Key mathematical concepts include the geometric properties and area calculation of ellipses. Using GeoGebra commands such as Intersect, Point, Circle, etc., dynamic exploration of the conditions for equal areas in an ellipse is possible. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Intersect(c, xAxis, 2) → C
  5. Segment(A, B) → f
  6. Point(f) → E
  7. Vector(A, E) → u
  8. Intersect(g, B1, (0, 0)) → C1
  9. Intersect(g, B10, (3.96181202750777, 2.1749487602193547)) → C10
  10. Intersect(g, B16, (7.7790211293602045, 3.6249146002958437)) → C16
  11. Intersect(g, B19, (8.944829003874156, 4.34989752034603)) → C19
  12. Intersect(g, B22, (9.904726917671706, 5.074880440428573)) → C22
  13. Intersect(g, B28, (13.464906929429503, 6.524846280532647)) → C28
  14. Intersect(g, B2, (0.8985363150627201, 0.2416609733505129)) → C2
  15. Intersect(g, B3, (1.4958358221658141, 0.48332194669894263)) → C3
  16. Intersect(g, B4, (1.9440988994429007, 0.7249829200567139)) → C4
  17. Intersect(g, B5, (2.321373279732653, 0.9666438934118669)) → C5
  18. Intersect(g, B6, (2.6616436981467055, 1.2083048678240824)) → C6
  19. Intersect(g, B7, (2.983486612768315, 1.4499658398307793)) → C7
  20. Intersect(g, B8, (3.299698691841183, 1.691626811800525)) → C8
  21. Intersect(g, B9, (3.6215416104738285, 1.9332877868253295)) → C9
  22. Intersect(g, B11, (4.339086409433638, 2.4166097345457658)) → C11
  23. Intersect(g, B12, (4.787349484998994, 2.658270706883755)) → C12
  24. Intersect(g, B13, (5.384648992109253, 2.899931680236732)) → C13
  25. Intersect(g, B14, (6.283185307198224, 3.1415926535941665)) → C14
  26. Intersect(g, B15, (7.181721622194075, 3.3832536269241604)) → C15
  27. Intersect(g, B17, (8.22728420171931, 3.8665755707204488)) → C17
  28. Intersect(g, B18, (8.604558586558609, 4.108236546760845)) → C18
  29. Intersect(g, B20, (9.266671920325276, 4.591558493708144)) → C20
  30. Intersect(g, B21, (9.58288399987242, 4.833219466039303)) → C21
  31. Intersect(g, B23, (10.24499733499598, 5.316541414019315)) → C23
  32. Intersect(g, B24, (10.622271717693026, 5.55820238877995)) → C24
  33. Intersect(g, B25, (11.07053478797479, 5.799863358455188)) → C25
  34. Intersect(g, B26, (11.667834299291737, 6.041524333827495)) → C26
  35. Intersect(g, B27, (12.56637, 6.283185163014739)) → C27
  36. Intersect(g, B29, (14.062206436539968, 6.766507253885722)) → C29
  37. Intersect(g, B30, (14.510469513806209, 7.008168227238768)) → C30
  38. Arc(c, (1; x(C1)), (1; x(C2))) → D2
  39. Arc(c, (1; x(C2)), (1; x(C3))) → D3
  40. Arc(c, (1; x(C3)), (1; x(C4))) → D4
  41. Arc(c, (1; x(C4)), (1; x(C5))) → D5
  42. Arc(c, (1; x(C5)), (1; x(C6))) → D6
  43. Arc(c, (1; x(C6)), (1; x(C7))) → D7
  44. Arc(c, (1; x(C7)), (1; x(C8))) → D8
  45. Arc(c, (1; x(C8)), (1; x(C9))) → D9
  46. Arc(c, (1; x(C9)), (1; x(C10))) → D10
  47. Arc(c, (1; x(C10)), (1; x(C11))) → D11
  48. Arc(c, (1; x(C11)), (1; x(C12))) → D12
  49. Arc(c, (1; x(C12)), (1; x(C13))) → D13
  50. Arc(c, (1; x(C13)), (1; x(C14))) → D14
  51. Arc(c, (1; x(C14)), (1; x(C15))) → D15
  52. Arc(c, (1; x(C15)), (1; x(C16))) → D16
  53. Arc(c, (1; x(C16)), (1; x(C17))) → D17
  54. Arc(c, (1; x(C17)), (1; x(C18))) → D18
  55. Arc(c, (1; x(C18)), (1; x(C19))) → D19
  56. Arc(c, (1; x(C19)), (1; x(C20))) → D20
  57. Arc(c, (1; x(C20)), (1; x(C21))) → D21
  58. Arc(c, (1; x(C21)), (1; x(C22))) → D22
  59. Arc(c, (1; x(C22)), (1; x(C23))) → D23
  60. Arc(c, (1; x(C23)), (1; x(C24))) → D24
  61. Arc(c, (1; x(C24)), (1; x(C25))) → D25
  62. Arc(c, (1; x(C25)), (1; x(C26))) → D26
  63. Arc(c, (1; x(C26)), (1; x(C27))) → D27
  64. Arc(c, (1; x(C27)), (1; x(C28))) → D28
  65. Arc(c, (1; x(C28)), (1; x(C29))) → D29
  66. Arc(c, (1; x(C29)), (1; x(C30))) → D30
  67. Segment((1; x(C1)), (1; x(C2))) → F2
  68. Segment((1; x(C2)), (1; x(C3))) → F3
  69. Segment((1; x(C3)), (1; x(C4))) → F4
  70. Segment((1; x(C4)), (1; x(C5))) → F5
  71. Segment((1; x(C5)), (1; x(C6))) → F6
  72. Segment((1; x(C6)), (1; x(C7))) → F7
  73. Segment((1; x(C7)), (1; x(C8))) → F8
  74. Segment((1; x(C8)), (1; x(C9))) → F9
  75. Segment((1; x(C9)), (1; x(C10))) → F10
  76. Segment((1; x(C10)), (1; x(C11))) → F11
  77. Segment((1; x(C11)), (1; x(C12))) → F12
  78. Segment((1; x(C12)), (1; x(C13))) → F13
  79. Segment((1; x(C13)), (1; x(C14))) → F14
  80. Segment((1; x(C14)), (1; x(C15))) → F15
  81. Segment((1; x(C15)), (1; x(C16))) → F16
  82. Segment((1; x(C16)), (1; x(C17))) → F17
  83. Segment((1; x(C17)), (1; x(C18))) → F18
  84. Segment((1; x(C18)), (1; x(C19))) → F19
  85. Segment((1; x(C19)), (1; x(C20))) → F20
  86. Segment((1; x(C20)), (1; x(C21))) → F21
  87. Segment((1; x(C21)), (1; x(C22))) → F22
  88. Segment((1; x(C22)), (1; x(C23))) → F23
  89. Segment((1; x(C23)), (1; x(C24))) → F24
  90. Segment((1; x(C24)), (1; x(C25))) → F25
  91. Segment((1; x(C25)), (1; x(C26))) → F26
  92. Segment((1; x(C26)), (1; x(C27))) → F27
  93. Segment((1; x(C27)), (1; x(C28))) → F28
  94. Segment((1; x(C28)), (1; x(C29))) → F29
  95. Segment((1; x(C29)), (1; x(C30))) → F30
  96. Vector(D, F) → v
  97. Invert(m) → mm
  98. ApplyMatrix(mm, D2) → J2
  99. ApplyMatrix(mm, D3) → J3
  100. ApplyMatrix(mm, D4) → J4
  101. ApplyMatrix(mm, D5) → J5
  102. ApplyMatrix(mm, D6) → J6
  103. ApplyMatrix(mm, D7) → J7
  104. ApplyMatrix(mm, D8) → J8
  105. ApplyMatrix(mm, D9) → J9
  106. ApplyMatrix(mm, D10) → J10
  107. ApplyMatrix(mm, D11) → J11
  108. ApplyMatrix(mm, D12) → J12
  109. ApplyMatrix(mm, D13) → J13
  110. ApplyMatrix(mm, D14) → J14
  111. ApplyMatrix(mm, D15) → J15
  112. ApplyMatrix(mm, D16) → J16
  113. ApplyMatrix(mm, D17) → J17
  114. ApplyMatrix(mm, D18) → J18
  115. ApplyMatrix(mm, D19) → J19
  116. ApplyMatrix(mm, D20) → J20
  117. ApplyMatrix(mm, D21) → J21
  118. ApplyMatrix(mm, D22) → J22
  119. ApplyMatrix(mm, D23) → J23
  120. ApplyMatrix(mm, D24) → J24
  121. ApplyMatrix(mm, D25) → J25
  122. ApplyMatrix(mm, D26) → J26
  123. ApplyMatrix(mm, D27) → J27
  124. ApplyMatrix(mm, D28) → J28
  125. ApplyMatrix(mm, D29) → J29
  126. ApplyMatrix(mm, D30) → J30
  127. ApplyMatrix(mm, E2) → K2
  128. ApplyMatrix(mm, F2) → L2
  129. ApplyMatrix(mm, E3) → K3
  130. ApplyMatrix(mm, F3) → L3
  131. ApplyMatrix(mm, E4) → K4
  132. ApplyMatrix(mm, F4) → L4
  133. ApplyMatrix(mm, E5) → K5
  134. ApplyMatrix(mm, F5) → L5
  135. ApplyMatrix(mm, E6) → K6
  136. ApplyMatrix(mm, F6) → L6
  137. ApplyMatrix(mm, E7) → K7
  138. ApplyMatrix(mm, F7) → L7
  139. ApplyMatrix(mm, E8) → K8
  140. ApplyMatrix(mm, F8) → L8
  141. ApplyMatrix(mm, E9) → K9
  142. ApplyMatrix(mm, F9) → L9
  143. ApplyMatrix(mm, E10) → K10
  144. ApplyMatrix(mm, F10) → L10
  145. ApplyMatrix(mm, E11) → K11
  146. ApplyMatrix(mm, F11) → L11
  147. ApplyMatrix(mm, E12) → K12
  148. ApplyMatrix(mm, F12) → L12
  149. ApplyMatrix(mm, E13) → K13
  150. ApplyMatrix(mm, F13) → L13
  151. ApplyMatrix(mm, E14) → K14
  152. ApplyMatrix(mm, F14) → L14
  153. ApplyMatrix(mm, E15) → K15
  154. ApplyMatrix(mm, F15) → L15
  155. ApplyMatrix(mm, E16) → K16
  156. ApplyMatrix(mm, F16) → L16
  157. ApplyMatrix(mm, E17) → K17
  158. ApplyMatrix(mm, F17) → L17
  159. ApplyMatrix(mm, E18) → K18
  160. ApplyMatrix(mm, F18) → L18
  161. ApplyMatrix(mm, E19) → K19
  162. ApplyMatrix(mm, F19) → L19
  163. ApplyMatrix(mm, E20) → K20
  164. ApplyMatrix(mm, F20) → L20
  165. ApplyMatrix(mm, E21) → K21
  166. ApplyMatrix(mm, F21) → L21
  167. ApplyMatrix(mm, E22) → K22
  168. ApplyMatrix(mm, F22) → L22
  169. ApplyMatrix(mm, E23) → K23
  170. ApplyMatrix(mm, F23) → L23
  171. ApplyMatrix(mm, E24) → K24
  172. ApplyMatrix(mm, F24) → L24
  173. ApplyMatrix(mm, E25) → K25
  174. ApplyMatrix(mm, F25) → L25
  175. ApplyMatrix(mm, E26) → K26
  176. ApplyMatrix(mm, F26) → L26
  177. ApplyMatrix(mm, E27) → K27
  178. ApplyMatrix(mm, F27) → L27
  179. ApplyMatrix(mm, E28) → K28
  180. ApplyMatrix(mm, F28) → L28
  181. ApplyMatrix(mm, E29) → K29
  182. ApplyMatrix(mm, F29) → L29
  183. ApplyMatrix(mm, E30) → K30
  184. ApplyMatrix(mm, F30) → L30

AI Drawing Prompt Reference

Draw an ellipse and select multiple points on it to construct several arcs, demonstrating the property of equal areas in an ellipse. First, select two points on the axes to form a circle with these points as the diameter. Then, select multiple points on the circle to construct several arcs.
The prompt above was compiled from the original drawing commands and is for reference only.