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Special Lines and Points in a Triangle

Special Lines and Points in a Triangle
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2016-02-22 21:28
TriangleCircleGeometric Construction
This example demonstrates the geometric properties and relationships of special line segments (like angle bisectors, medians, altitudes) and special points (like incenter, circumcenter, centroid, orthocenter) in a triangle. Key math concepts include angle bisector theorem, median properties, centroid and orthocenter coordinates, and the relationship between incenter and circumcenter. Construction highlights involve GeoGebra commands such as Polygon for triangle creation, LineBisector for angle bisectors, Intersect for intersection points, AngularBisector for angular bisectors, Segment for line segments, Circle for drawing circles, OrthogonalLine for perpendicular lines, and Angle for angle measurement. Suitable for high school geometry construction teaching and exploratory learning, helping students understand the properties and connections of triangle centers and special segments.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. LineBisector(c) → pbc
  3. LineBisector(a) → pba
  4. LineBisector(b) → pbb
  5. Intersect(pbc, pba) → CC
  6. AngularBisector(B, A, C) → aba
  7. AngularBisector(A, C, B) → abc
  8. AngularBisector(A, B, C) → abb
  9. Intersect(aba, abc) → CI
  10. Intersect(pba, a) → MA
  11. Intersect(pbb, b) → MB
  12. Intersect(pbc, c) → MC
  13. Segment(A, MA) → ma
  14. Segment(B, MB) → mb
  15. Segment(C, MC) → mc
  16. Intersect(ma, mb) → CG
  17. Circle(CC, B) → cc
  18. OrthogonalLine(CI, a) → auxlineperp
  19. Intersect(auxlineperp, a) → AUXpoint
  20. Circle(CI, AUXpoint) → ci
  21. Angle(C, MA, CC) → α
  22. Angle(B, MC, CC) → β
  23. Angle(A, MB, CC) → γ
  24. Angle(C, B, CI) → δ
  25. Angle(CI, B, A) → ε
  26. Angle(B, A, CI) → ζ
  27. Angle(CI, A, C) → η
  28. Angle(A, C, CI) → θ
  29. Angle(CI, C, B) → ι
  30. Segment(C, MA) → a1
  31. Segment(MA, B) → a2
  32. Segment(A, MB) → b1
  33. Segment(MB, C) → b2
  34. Segment(B, MC) → c1
  35. Segment(MC, A) → c2
  36. Angle(B, MC, SC) → κ
  37. Angle(A, MB, SB) → λ
  38. Angle(C, MA, SA) → μ
  39. OrthogonalLine(A, a) → hlineA
  40. OrthogonalLine(B, b) → hlineB
  41. OrthogonalLine(C, c) → hlineC
  42. Intersect(hlineA, Line[B, C]) → fA
  43. Intersect(hlineB, Line[A, C]) → fB
  44. Intersect(hlineC, Line[A, B]) → fC
  45. Intersect(hlineA, hlineB) → ORT
  46. Segment(fA, B) → fAB
  47. Segment(fA, C) → fAC
  48. Segment(fB, A) → fBA
  49. Segment(fB, C) → fBC
  50. Segment(fC, A) → fCA
  51. Segment(fC, B) → fCB
  52. Intersect(fCB, fAB) → E
  53. Segment(A, fA) → hA
  54. Segment(B, fB) → hB
  55. Segment(C, fC) → hC
  56. Segment(fA, ORT) → auxhA
  57. Segment(fB, ORT) → auxhB
  58. Segment(fC, ORT) → auxhC
  59. Angle(C, fA, A) → angfA1
  60. Angle(B, fC, C) → angfC1
  61. Angle(A, fB, B) → angfB1
  62. Angle(A, fA, C) → angfA2
  63. Angle(B, fB, A) → angfB2
  64. Angle(C, fC, B) → angfC2
  65. Polygon(Corner[1], Corner[2], Corner[3], Corner[4]) → poly2, d, e, f, g