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Demonstration of Triangle's 4 Concurrency Points

Demonstration of Triangle's 4 Concurrency Points
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2019-11-14 02:36
TriangleCircleGeometric Construction
This example demonstrates the concurrency of four significant points in a triangle: centroid, orthocenter, incenter, and circumcenter. It involves core mathematical concepts such as triangle geometry, the theorem of four centers, and analytical geometry constructions. The construction highlights GeoGebra commands like Line, Segment, OrthogonalLine, Intersect, Midpoint, Angle, AngularBisector, and LineBisector, used to build triangle vertices, sides, altitudes, medians, angle bisectors, and circles. It is suitable for high school and above students to understand key geometric properties and construction techniques of triangles.

GeoGebra Commands & Parameters

  1. Line(A, B) → LineAB
  2. Line(B, C) → LineBC
  3. Line(C, A) → SegAC
  4. Segment(B, A) → SegAB
  5. Segment(A, C) → SegAC_1
  6. Segment(C, B) → SegBC
  7. OrthogonalLine(A, LineBC) → f
  8. OrthogonalLine(C, LineAB) → g
  9. OrthogonalLine(B, SegAC) → h
  10. Intersect(SegAC, h) → D
  11. Intersect(g, LineAB) → F
  12. Intersect(LineBC, f) → G
  13. Segment(B, D) → AltitudeBD
  14. Segment(A, G) → AltitudeAG
  15. Segment(F, C) → AltitudeFC
  16. Midpoint(B, C) → H
  17. Midpoint(B, A) → I
  18. Midpoint(A, C) → J
  19. Segment(I, C) → MedianCI
  20. Segment(B, J) → MedianBJ
  21. Segment(A, H) → MedianAH
  22. Angle(A, G, B) → α
  23. Angle(B, D, C) → β
  24. Angle(C, F, A) → γ
  25. Segment(A, I) → AI
  26. Segment(I, B) → BI
  27. Segment(B, H) → BH
  28. Segment(H, C) → CH
  29. Segment(A, J) → AJ
  30. Segment(J, C) → CJ
  31. Intersect(MedianCI, MedianBJ) → E
  32. Intersect(AltitudeBD, AltitudeAG) → K
  33. AngularBisector(B, C, A) → i
  34. AngularBisector(C, A, B) → j
  35. AngularBisector(A, B, C) → k
  36. Intersect(j, LineBC) → L
  37. Intersect(LineAB, i) → M
  38. Intersect(k, SegAC) → P
  39. Segment(B, P) → AngBisBP
  40. Segment(C, M) → AngBisCM
  41. Segment(L, A) → AngBisAL
  42. Intersect(AngBisBP, AngBisCM) → InCenterN
  43. LineBisector(SegAC_1) → l
  44. LineBisector(SegAB) → m
  45. LineBisector(SegBC) → n
  46. Intersect(LineBC, n) → N
  47. Intersect(LineAB, m) → O
  48. Intersect(SegAC, l) → Q
  49. Intersect(l, m) → R
  50. Segment(R, Q) → PerpBisPQ
  51. Segment(R, N) → PerpBisRN
  52. Segment(R, O) → PerpBisOR
  53. Angle(R, Q, C) → δ
  54. Angle(R, N, B) → ε
  55. Angle(R, O, A) → ζ
  56. Line(N, R) → p
  57. Line(Q, R) → q
  58. Line(R, O) → r