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Beyond the Median

Beyond the Median
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2014-01-27 20:18
TrianglePolygonGeometric Construction
In collections Functions on the Domain of Space - Polygons
This applet demonstrates the generalization of the median concept in Euclidean geometry, involving triangles and polygons. It uses commands like Polygon, Circle, Intersect, Segment, If, and Distance, suitable for high school and above students to explore geometric transformations and properties of medians.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Circle(A, Distance[A, B] / f) → d
  3. Circle(B, Distance[B, C] / f) → e
  4. Circle(C, Distance[C, A] / f) → g
  5. Intersect(d, c) → D,
  6. Intersect(e, a) → E,
  7. Intersect(g, b) → F,
  8. Segment(A, E) → h
  9. Segment(B, F) → i
  10. Segment(C, D) → j
  11. Intersect(h, j) → G
  12. Intersect(i, h) → H
  13. Intersect(j, i) → I
  14. Polygon(G, H, I) → poly2, i_1, g_1, h_1
  15. If(f ≠ 2, Area[poly1] / Area[poly2]) → B1
  16. Segment(C, F) → l
  17. Segment(A, D) → m
  18. Segment(B, E) → n
  19. Distance(A, G) → AG
  20. Distance(A, E) → AE
  21. Distance(G, E) → GE
  22. Distance(G, H) → GH
  23. Distance(B, H) → BH
  24. Distance(H, I) → HI
  25. Distance(C, I) → CI
  26. Distance(I, G) → IG
  27. If(f ≠ 2, BH / HI) → B16
  28. If(f ≠ 2, AG / GH) → B15
  29. If(f ≠ 2, CI / IG) → B17

AI Drawing Prompt Reference

Draw a triangle ABC, then draw circles with centers at the midpoints of AB, BC, and CA, and radii equal to the distances from the midpoints to the vertices. Find the intersection points D, E, and F of these circles. Connect AD, BE, and CF, and find the intersection points G, H, and I of these lines. Finally, draw triangle GHI. Note that when f is not equal to 2, calculate and display the ratio of the area of triangle ABC to the area of triangle GHI.
The prompt above was compiled from the original drawing commands and is for reference only.