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Orthocenter Divides Altitudes to Create Equal Products

Orthocenter Divides Altitudes to Create Equal Products
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2016-02-28 19:27
TriangleCoordinate GeometryGeometric Construction
This example demonstrates how the orthocenter of a triangle divides altitudes to create equal products. Key mathematical concepts include the properties of altitudes and orthocenters. Using GeoGebra commands such as Polygon, OrthogonalLine, Intersect, Segment, Rotate, and PointIn, a dynamic geometric figure is constructed. Suitable for middle school and above students to learn triangle geometry.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. OrthogonalLine(A, a) → d
  3. OrthogonalLine(B, b) → e
  4. OrthogonalLine(C, c) → f
  5. Intersect(d, e) → D
  6. Segment(B, D) → g
  7. Segment(D, C) → h
  8. Segment(D, A) → i
  9. Intersect(e, b) → E
  10. Segment(B, E) → j
  11. Intersect(d, a) → F
  12. Segment(A, F) → k
  13. Intersect(f, c) → G
  14. Segment(C, G) → l
  15. Segment(D, A) → m
  16. Segment(D, A) → n
  17. Segment(D, F) → p
  18. Segment(B, D) → q
  19. Segment(E, D) → r
  20. Segment(G, D) → s
  21. Segment(D, C) → t
  22. Rotate(E, 90°, D) → E'
  23. Rotate(D, 90°, D) → D'
  24. Segment(E', D') → r'
  25. Rotate(G, 90°, D) → G'
  26. Rotate(D, 90°, D) → D'_1
  27. Segment(G', D'_1) → s'
  28. Rotate(D, 90°, D) → D'_2
  29. Rotate(F, 90°, D) → F'
  30. Segment(D'_2, F') → p'
  31. OrthogonalLine(E', r') → a_1
  32. OrthogonalLine(B, q) → b_1
  33. Intersect(a_1, b_1) → H
  34. Polygon(H, B, D, E') → poly2, h_1, b_2, d_1, e'
  35. OrthogonalLine(G', s') → c_1
  36. OrthogonalLine(C, t) → e_1
  37. Intersect(c_1, e_1) → I
  38. Polygon(D, C, I, G') → poly3, d_2, c_2, i_1, g'
  39. OrthogonalLine(F', p') → f_1
  40. OrthogonalLine(A, m) → g_1
  41. Intersect(f_1, g_1) → J
  42. Polygon(J, A, D, F') → poly4, j_1, a_2, d_3, f'
  43. PointIn(poly2) → Pointpoly2
  44. PointIn(poly4) → Pointpoly4
  45. PointIn(poly3) → Pointpoly3

AI Drawing Prompt Reference

Draw a triangle ABC, then drop perpendiculars from A, B, and C to the opposite sides, intersecting at points D, E, and F. Connect DE, DF, and EF, and observe how D divides these segments such that DE·DF = BE·EF = CF·CD. Then, rotate the perpendiculars around point D to form new triangles and observe how the perpendicular intersections H, I, and J relate to the original triangle's vertices.
The prompt above was compiled from the original drawing commands and is for reference only.