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Triangle Concurrency Demo

Triangle Concurrency Demo
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2020-11-13 19:29
TriangleCircleGeometric Construction
This example demonstrates key points of a triangle such as centroid, orthocenter, circumcenter, and incenter, showcasing their concurrencies and relationships. Core math concepts include triangle centers, altitudes, medians, angle bisectors, circumcircle, incircle, and geometric constructions. It is constructed using commands like Polygon, Midpoint, Intersect, LineBisector, Circle, etc. Ideal for high school and advanced learners exploring triangle geometry and constructions.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Midpoint(c) → M_C
  3. Midpoint(a) → M_A
  4. Midpoint(b) → M_B
  5. Segment(M_C, C) → f
  6. Segment(B, M_B) → g
  7. Segment(A, M_A) → h
  8. Intersect(f, g) → centroid
  9. LineBisector(c) → j
  10. LineBisector(a) → k
  11. LineBisector(b) → l
  12. Intersect(j, k) → circumcentre
  13. Angle(C, M_B, circumcentre) → α
  14. Angle(A, M_C, circumcentre) → β
  15. Angle(B, M_A, circumcentre) → γ
  16. Segment(A, M_C) → m
  17. Segment(M_C, B) → n
  18. Segment(A, M_B) → p
  19. Segment(M_B, C) → q
  20. Segment(C, M_A) → r
  21. Segment(M_A, B) → s
  22. AngularBisector(C, A, B) → f_1
  23. AngularBisector(A, B, C) → g_1
  24. AngularBisector(B, C, A) → h_1
  25. Intersect(h_1, f_1) → incentre
  26. Angle(C, A, incentre) → δ
  27. Angle(incentre, A, B) → ε
  28. Angle(A, B, incentre) → ζ
  29. Angle(incentre, B, C) → η
  30. Angle(B, C, incentre) → θ
  31. Angle(incentre, C, A) → ι
  32. OrthogonalLine(incentre, c) → i_1
  33. Intersect(i_1, c) → D
  34. Circle(incentre, D) → incircle
  35. Circle(A, B, C) → d_1
  36. Angle(A, D, incentre) → κ
  37. Segment(circumcentre, B) → r_1
  38. Segment(circumcentre, A) → r_2
  39. Segment(circumcentre, C) → r_3
  40. OrthogonalLine(C, c) → alt_1
  41. OrthogonalLine(A, a) → alt_2
  42. Intersect(alt_1, c) → F
  43. Intersect(a, alt_2) → G
  44. Distance(circumcentre, centroid) → distancecircumcentrecentroid
  45. OrthogonalLine(B, b) → alt_3
  46. Intersect(b, alt_3) → E
  47. Angle(B, E, A) → λ
  48. Intersect(alt_1, alt_3) → orthocentre
  49. Angle(A, F, orthocentre) → μ
  50. Angle(B, G, orthocentre) → ν
  51. Segment(circumcentre, centroid) → j_1
  52. Segment(centroid, orthocentre) → k_1

AI Drawing Prompt Reference

Draw an arbitrary triangle with three vertices. Connect each midpoint of a side to the opposite vertex to form three medians, intersecting at the centroid. Construct the perpendicular bisectors of all three sides, meeting at the circumcenter, and draw the circumcircle centered there passing through all vertices. Construct the three angle bisectors, meeting at the incenter, and draw the incircle centered there tangent to all sides. Drop perpendiculars from each vertex to the opposite side; these three altitudes intersect at the orthocenter. Finally, connect the circumcenter to each vertex with segments, and label the centroid, circumcenter, incenter, orthocenter, and relevant angles.
The prompt above was compiled from the original drawing commands and is for reference only.