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The Nine-Point Circle and the Euler Line

The Nine-Point Circle and the Euler Line
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2015-02-14 05:53
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the nine-point circle and Euler line in a triangle. Key mathematical concepts include the pedal triangle, centroid, orthocenter, and circumcenter. Construction involves GeoGebra commands such as Line, OrthogonalLine, Intersect, LineBisector, Mirror, Circle, Midpoint, and Segment. Suitable for middle or high school geometry learning to understand basic triangle properties and geometric relationships.

GeoGebra Commands & Parameters

  1. Line(A, B) → m
  2. Line(C, A) → n
  3. Line(B, C) → q
  4. OrthogonalLine(A, q) → a
  5. OrthogonalLine(B, n) → b
  6. OrthogonalLine(C, m) → c
  7. Intersect(a, b) → H
  8. LineBisector(B, C) → d
  9. LineBisector(C, A) → e
  10. LineBisector(A, B) → f
  11. Intersect(d, e) → O
  12. Mirror(O, n) → O'_3
  13. Mirror(O, q) → O'_1
  14. Mirror(O, m) → O'_2
  15. Mirror(H, m) → H'_3
  16. Mirror(H, n) → H'_1
  17. Mirror(H, q) → H'_2
  18. Circle(H'_3, H'_1, H'_2) → g
  19. Circle(O'_2, O'_3, O'_1) → h
  20. Intersect(q, a) → D
  21. Intersect(n, e) → E
  22. Intersect(n, b) → F
  23. Intersect(m, c) → G_1
  24. Intersect(m, f) → I
  25. Intersect(q, d) → J
  26. Circle(G_1, D, E) → k
  27. Midpoint(H, O) → N
  28. Segment(A, J) → i
  29. Segment(B, E) → j
  30. Segment(I, C) → l
  31. Intersect(i, j) → G
  32. Line(H, O) → r
  33. Mirror(H, O) → L

AI Drawing Prompt Reference

Draw a triangle. Construct the three altitudes, which meet at the orthocenter. Then construct the perpendicular bisectors of the three sides, which meet at the circumcenter. Reflect the circumcenter across each side to get three points, and draw the circle through them. Similarly, reflect the orthocenter across each side to get three points, and draw the circle through them. Also, find the intersections of each side with its altitude and perpendicular bisector, and draw a circle through three of these points. Connect the orthocenter and circumcenter, take their midpoint as the center of the nine-point circle. Finally, connect each vertex to the midpoint of the opposite side; the three medians meet at the centroid, and draw the line through the orthocenter and circumcenter.
The prompt above was compiled from the original drawing commands and is for reference only.