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Euler Line of a Triangle

Euler Line of a Triangle
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2016-08-02 18:24
TriangleCoordinate GeometryGeometric Construction
This example demonstrates the construction of the orthocenter, centroid, and circumcenter of any triangle, and adds the Euler line. By measuring the ratios in associated triangles, important geometric concepts are explored. Commands used include Segment, LineBisector, Intersect, Line, OrthogonalLine, Angle, Polygon, Midpoint, etc. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Segment(C, B) → f
  2. Segment(A, B) → g
  3. Segment(A, C) → h
  4. LineBisector(h) → cc1
  5. LineBisector(f) → cc2
  6. LineBisector(g) → cc3
  7. Intersect(cc1, cc2) → Circumcentre
  8. Intersect(cc2, f) → E_1
  9. Intersect(cc1, h) → F
  10. Intersect(cc3, g) → G_1
  11. Segment(C, G_1) → cen1
  12. Segment(B, F) → cwn2
  13. Segment(A, E_1) → n
  14. Intersect(cen1, cwn2) → Centroid
  15. Line(A, B) → o4
  16. Line(B, C) → o5
  17. Line(C, A) → o6
  18. OrthogonalLine(C, g) → o2
  19. OrthogonalLine(A, f) → o3
  20. OrthogonalLine(B, h) → o1
  21. Intersect(o2, o3) → Orthocentre
  22. Segment(Orthocentre, Circumcentre) → E
  23. Angle(h, cc1) → cc4
  24. Angle(g, cc3) → cc5
  25. Angle(f, cc2) → cc6
  26. Angle(h, o1) → o7
  27. Angle(g, o2) → o8
  28. Angle(f, o3) → o9
  29. Angle(B, A, C) → η
  30. Angle(C, B, A) → θ
  31. Angle(A, C, B) → ι
  32. Segment(Centroid, Circumcentre) → cc
  33. Segment(Orthocentre, Centroid) → oc
  34. Segment(Circumcentre, E_1) → d
  35. Intersect(o3, f) → D
  36. Segment(Orthocentre, D) → f_1
  37. Intersect(n, d) → G
  38. Polygon(Circumcentre, Centroid, G) → bluetri, g_1, circumcentre, centroid
  39. Polygon(Centroid, A, Orthocentre) → yelltri, orthocentre, centroid_1, a_1
  40. Midpoint(B, C) → H
  41. Segment(H, A) → i

AI Drawing Prompt Reference

Draw an arbitrary triangle with three vertices. Construct the perpendicular bisectors of all three sides; they intersect at the circumcenter. Draw the three medians; they meet at the centroid. Draw the three altitudes; they intersect at the orthocenter. Connect the circumcenter, centroid, and orthocenter to form the Euler line. Also draw segments from the circumcenter to the midpoints of the sides, and from the orthocenter to the feet of the altitudes. Mark the three interior angles of the triangle. Highlight two triangles: one formed by the circumcenter, centroid, and orthocenter, and another formed by the centroid, one vertex, and the orthocenter, filling them with different colors. Finally, take the midpoint of one side and connect it to the opposite vertex to show a median.
The prompt above was compiled from the original drawing commands and is for reference only.