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Centers of a Triangle by Clio Hartzer

Centers of a Triangle by Clio Hartzer
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2014-04-13 05:24
TriangleCircleAngleVectorGeometric Construction
This applet demonstrates the four basic 'centers' of a triangle (centroid, circumcenter, incenter, orthocenter) and their relationships, along with midsegments. Constructed using GeoGebra commands like Segment, AngularBisector, Line, OrthogonalLine, LineBisector, Intersect, Circle, and Angle, it involves points, sides, angle bisectors, altitudes, perpendicular bisectors, circumcircle, incircle, etc., making it suitable for high school geometry education and construction learning.

GeoGebra Commands & Parameters

  1. Segment(PointC, PointB) → SideA
  2. Segment(PointA, PointC) → SideB
  3. Segment(PointB, PointA) → SideC
  4. AngularBisector(PointA, PointC, PointB) → AngleBisLineB2
  5. Line(PointC, PointB) → LineA
  6. OrthogonalLine(PointA, LineA) → AltitudeB1
  7. Line(PointA, PointC) → LineB
  8. OrthogonalLine(PointB, LineB) → AltitudeB3
  9. Line(PointB, PointA) → LineC
  10. OrthogonalLine(PointC, LineC) → AltitudeB2
  11. LineBisector(SideC) → PerpBisB1
  12. LineBisector(SideB) → PerpBisB3
  13. LineBisector(SideA) → PerpBisB2
  14. Intersect(PerpBisB1, PerpBisB2) → Circumcenter
  15. Segment(Circumcenter, PointC) → Circum1
  16. Segment(Circumcenter, PointA) → Circum2
  17. Segment(Circumcenter, PointB) → Circum3
  18. Circle(Circumcenter, Circum1) → Circumcircle
  19. Intersect(PerpBisB2, SideA) → Midpoint1
  20. Intersect(PerpBisB1, SideC) → Midpoint2
  21. Intersect(PerpBisB3, SideB) → Midpoint3
  22. Segment(Circumcenter, Midpoint3) → PerpBis1
  23. Segment(Circumcenter, Midpoint1) → PerpBis2
  24. Segment(Circumcenter, Midpoint2) → PerpBis3
  25. AngularBisector(PointB, PointA, PointC) → AngleBisLineB1
  26. AngularBisector(PointC, PointB, PointA) → AngleBisLineB3
  27. Intersect(AngleBisLineB1, AngleBisLineB2) → IncenterLine
  28. OrthogonalLine(IncenterLine, LineC) → IncenterB1
  29. OrthogonalLine(IncenterLine, LineA) → IncenterB2
  30. OrthogonalLine(IncenterLine, LineB) → IncenterB3
  31. Intersect(IncenterB1, SideC) → IncenterA1
  32. Intersect(IncenterB3, SideB) → IncenterA2
  33. Intersect(IncenterB2, SideA) → IncenterA3
  34. Intersect(AngleBisLineB1, SideA) → AngBisA1
  35. Intersect(AngleBisLineB3, SideB) → AngBisA2
  36. Segment(AngBisA1, PointA) → AngBisect1
  37. Intersect(AngleBisLineB2, SideC) → AngBisA3
  38. Segment(AngBisA3, PointC) → AngBisect2
  39. Segment(AngBisA2, PointB) → AngBisect3
  40. Segment(IncenterLine, IncenterA1) → Incenter1
  41. Segment(IncenterLine, IncenterA3) → Incenter2
  42. Segment(IncenterLine, IncenterA2) → Incenter3
  43. Intersect(AngBisect1, AngBisect2) → Incenter
  44. Circle(Incenter, Incenter1) → Incircle
  45. Angle(Incenter, IncenterA2, PointA) → IncenterA
  46. Angle(Incenter, IncenterA1, PointB) → IncenterB
  47. Angle(Incenter, IncenterA3, PointC) → IncenterC
  48. Segment(PointB, Midpoint3) → Median1
  49. Segment(Midpoint2, PointC) → Median2
  50. Segment(PointA, Midpoint1) → Median3
  51. Intersect(Median1, Median2) → Centroid
  52. Intersect(AltitudeB1, AltitudeB2) → Orthocenter
  53. Intersect(AltitudeB2, LineC) → AltitudeA1
  54. Angle(Orthocenter, AltitudeA1, PointB) → AltitudeA
  55. Intersect(AltitudeB1, LineA) → AltitudeA3
  56. Angle(Orthocenter, AltitudeA3, PointC) → AltitudeB
  57. Intersect(AltitudeB3, LineB) → AltitudeA2
  58. Angle(Orthocenter, AltitudeA2, PointA) → AltitudeC
  59. Segment(PointC, AltitudeA1) → Altitude1
  60. Segment(PointA, AltitudeA3) → Altitude2
  61. Segment(PointB, AltitudeA2) → Altitude3
  62. Angle(Circumcenter, Midpoint2, PointB) → PerpBisA
  63. Angle(Circumcenter, Midpoint1, PointC) → PerpBisB
  64. Angle(Circumcenter, Midpoint3, PointA) → PerpBisC
  65. Segment(Midpoint2, PointA) → MidpointC1
  66. Segment(Midpoint2, PointB) → MidpointC2
  67. Segment(PointB, Midpoint1) → MidpointA1
  68. Segment(Midpoint1, PointC) → MidpointA2
  69. Segment(PointC, Midpoint3) → MidpointB1
  70. Segment(Midpoint3, PointA) → MidpointB2
  71. Angle(AngBisA3, PointB, AngBisA2) → AngleBisA1
  72. Angle(AngBisA2, PointB, PointC) → AngleBisA2
  73. Angle(PointC, PointA, AngBisA1) → AngleBisB1
  74. Angle(AngBisA1, PointA, PointB) → AngleBisB2
  75. Angle(PointB, PointC, AngBisA3) → AngleBisC1
  76. Angle(AngBisA3, PointC, PointA) → AngleBisC2
  77. Segment(Midpoint2, Midpoint1) → MidB
  78. Segment(Midpoint1, Midpoint3) → MidC
  79. Segment(Midpoint3, Midpoint2) → MidA
  80. OrthogonalLine(PointA, LineA) → AltLineA
  81. OrthogonalLine(PointC, LineC) → AltLineC
  82. OrthogonalLine(PointB, LineB) → AltLineB
  83. Line(Circumcenter, Orthocenter) → EulerLine
  84. Angle(PointA, PointC, PointB) → VertC
  85. Angle(PointC, PointB, PointA) → VertA
  86. Angle(PointB, PointA, PointC) → VertB

AI Drawing Prompt Reference

Draw a triangle and construct its three perpendicular bisectors of sides, their intersection is the circumcenter. Connect the circumcenter to the vertices to form the circumcircle. Then, draw the three angle bisectors, their intersection is the incenter. Connect the incenter to the sides to form the incircle. Also, draw the three altitudes, their intersection is the orthocenter. Finally, connect the midpoints of the sides to show the midsegments and relationships among these centers and segments.
The prompt above was compiled from the original drawing commands and is for reference only.