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Archimedes and Parallelogram's Center of Gravity

Archimedes and Parallelogram's Center of Gravity
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2017-10-08 02:15
PolygonCoordinate GeometryGeometric Construction
This example demonstrates Archimedes' theorem on the center of gravity of a parallelogram. Key mathematical concepts include parallelogram, center of gravity, midpoint, and line. The construction uses GeoGebra commands such as Point, Segment, Midpoint, and Line. Suitable for middle school and above students to learn geometric construction and related theorems.

GeoGebra Commands & Parameters

  1. Point(Ray[C, C - (1, 0)]) → B
  2. Segment(A, D) → f
  3. Segment(D, C) → g
  4. Segment(C, B) → h
  5. Segment(B, A) → i
  6. Midpoint(f) → E
  7. Midpoint(h) → F
  8. Segment(E, F) → j
  9. Line(H, f) → k
  10. Intersect(k, j) → K
  11. Segment(K, H) → l
  12. Distance(E, D) → ED
  13. Distance(K, H) → KH
  14. Sequence(E + ((k * a), 0), k, 1, ceil(ED / KH)) → L_1
  15. Sequence(F + ((k * a), 0), k, 1, ceil(ED / KH)) → L_2
  16. Sequence(Segment[Element[L_1, k], Element[L_2, k]], k, 1, ceil(ED / KH)) → L_3
  17. Sequence(E - ((k * a), 0), k, 1, ceil(ED / KH)) → L_4
  18. Sequence(F - ((k * a), 0), k, 1, ceil(ED / KH)) → L_5
  19. Sequence(Segment[Element[L_4, k], Element[L_5, k]], k, 1, ceil(ED / KH)) → L_6
  20. Distance(A, D) → AD
  21. Distance(A, B) → AB
  22. Intersect(k, i) → J
  23. Distance(A, J) → AJ
  24. Element(Last[L_4], 1) → L
  25. Sequence(G + (k * Vector[A, D] / ((2 * c))), k, 1, (2 * c) - 1) → L_7
  26. Segment(Element[L_7, c - 1], Element[L_7, c]) → m
  27. Midpoint(m) → I

AI Drawing Prompt Reference

Draw a parallelogram ABCD, connect its diagonals AC and BD to find their midpoints E and F, then draw a line through E and F. Next, find the intersection K of these diagonals, draw a line through K and E, and another through K and F. Then, draw a line segment through E and F, and draw parallel lines through E and F until they intersect with AB and CD. Finally, draw a line segment through A and D, find its intersection J with BC, and draw a line segment through the midpoint I of EF.
The prompt above was compiled from the original drawing commands and is for reference only.