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Opposite Sides Equal Implies Parallelogram Demo

Opposite Sides Equal Implies Parallelogram Demo
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2020-08-29 01:46
PolygonVectorTransformationGeometric Construction
This example demonstrates that a quadrilateral with equal opposite sides must be a parallelogram. Core concepts include quadrilateral properties, the parallelogram criterion, segment translation and congruence, and vector parallelism. Construction uses GeoGebra commands Segment, Line, Intersect, Circle, Arc, Translate, and conditional If; objects comprise vertices A, B, C, D and associated segments a, b, c, d, f, g, h, i, j, k, l, m, n, p, q, r, s. Suitable for middle‑school geometry (grade 7) lessons on parallelogram identification, high‑school review, or introductory GeoGebra construction workshops.

GeoGebra Commands & Parameters

  1. Segment(A, C) → f
  2. Segment(A, B) → g
  3. If(a < 1, Segment[Translate[A, Vector[(a * Vector[A, C])]], Translate[B, Vector[(a * Vector[A, C])]]], Segment[Translate[A, Vector[A, C]], Translate[B, Vector[A, C]]]) → h
  4. Line(C, g) → i
  5. Line(B, f) → j
  6. Intersect(i, j) → D
  7. Circle(C, D) → c
  8. Arc(c, D, Rotate[D, (((a - 1) * 2) * π), C]) → d
  9. If(2 < a ≤ 3, Segment[Translate[A, Vector[((a - 2) * Vector[A, B])]], Translate[C, Vector[((a - 2) * Vector[A, B])]]], Segment[B, D]) → k
  10. Circle(B, D) → e
  11. Arc(e, D, Rotate[D, (((a - 3) * 2) * π), B]) → p
  12. Segment(B, Translate[B, Vector[((a - 4.5) * Vector[B, D])]]) → l
  13. Segment(B, D) → m
  14. Segment(C, D) → n
  15. Segment(C, Translate[C, Vector[((a - 4.5) * Vector[C, D])]]) → q
  16. Translate(Line[A, C], Vector[((a - 5.5) / 0.75 * Vector[A, B])]) → r
  17. Translate(Line[A, B], Vector[((a - 6.25) / 0.75 * Vector[A, C])]) → s

AI Drawing Prompt Reference

Draw a quadrilateral with equal opposite sides to demonstrate it must be a parallelogram. Start by creating two equal segments and conditionally translating them to form the fourth vertex of the quadrilateral. Use circles and arcs to further confirm the equality of opposite sides, and draw the associated lines and arcs. Finally, translate the sides and diagonals to show that the opposite sides are parallel and equal, proving the quadrilateral is a parallelogram.
The prompt above was compiled from the original drawing commands and is for reference only.