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Centroid, Circumcenter, Incenter and Orthocenter

Centroid, Circumcenter, Incenter and Orthocenter
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2017-12-09 11:45
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the centroid, circumcenter, incenter, and orthocenter of a triangle. Key mathematical concepts include the geometric properties and theorems of triangles. Using GeoGebra commands such as Polygon, Segment, Intersect, Circle, etc., it provides a visual understanding of these important geometric concepts. Suitable for middle school and above students to learn geometric construction and related theorems.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → t1, c, a, b
  2. Segment(Midpoint[a], A) → r
  3. Segment(Midpoint[c], C) → p
  4. Intersect(LineBisector[a], LineBisector[b]) → Circumcenter
  5. Segment(Midpoint[c], Circumcenter) → i
  6. Segment(Midpoint[b], Circumcenter) → j
  7. Segment(Midpoint[a], Circumcenter) → k
  8. Segment(A, Circumcenter) → l
  9. Segment(Circumcenter, C) → m
  10. Segment(Circumcenter, B) → n
  11. Segment(Midpoint[b], B) → q
  12. Intersect(p, r) → Centroid
  13. Segment(B, Centroid) → s
  14. Segment(A, Centroid) → t
  15. Segment(C, Centroid) → u
  16. Circle(A, B, C) → d
  17. Segment(C, Center[Incircle[A, B, C]]) → gi
  18. Segment(A, Center[Incircle[A, B, C]]) → hi
  19. Intersect(gi, hi) → Incenter
  20. Incircle(A, B, C) → Incircle
  21. Segment(B, Center[Incircle[A, B, C]]) → fi
  22. Segment(Incenter, Intersect[OrthogonalLine[Incenter, a], a]) → f
  23. Segment(Incenter, Intersect[OrthogonalLine[Incenter, b], b]) → g
  24. Segment(Incenter, Intersect[OrthogonalLine[Incenter, c], c]) → h
  25. OrthogonalLine(C, Line[B, A]) → g_1
  26. OrthogonalLine(A, Line[B, C]) → h_1
  27. Intersect(g_1, h_1) → Orthocenter
  28. OrthogonalLine(B, Line[C, A]) → f_1
  29. Line(B, c) → i_1
  30. Line(C, b) → j_1
  31. Line(C, a) → k_1

AI Drawing Prompt Reference

Draw a triangle ABC, then find and draw its circumcenter, incenter, and orthocenter. First, draw the triangle, then find the midpoints of each side and draw the perpendicular bisectors through the midpoints and vertices, with their intersection being the circumcenter. Next, draw the incircle, and the intersection of the perpendicular bisectors with the sides will give the incenter. Finally, draw perpendicular lines from the incenter to each side, and their intersection will be the orthocenter.
The prompt above was compiled from the original drawing commands and is for reference only.