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Steiner Point

Steiner Point
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2015-02-16 21:12
TriangleCircleGeometric Construction
In collections A Tour of Triangle Geometry
This example demonstrates the Steiner point and its relationship with a triangle. Key mathematical concepts include inversion, concyclic points, and triangle geometry. Construction involves commands like Line, Circle, Centroid, Intersect, and Conic to create inversion points, circles, and intersection points. Suitable for high school students and geometry enthusiasts studying triangle geometry and transformations.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. Line(B, C) → b
  3. Line(C, A) → c
  4. Circle(A, B, C) → d
  5. Centroid(Polygon[A, B, C]) → G
  6. OrthogonalLine(A, b) → e
  7. OrthogonalLine(B, c) → f
  8. Intersect(e, f) → H
  9. Center(d) → O
  10. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  11. Mirror(A, G) → A'
  12. Mirror(B, G) → B'
  13. Mirror(C, G) → C'
  14. Conic(C', A, B', C, B) → g
  15. Mirror(G, b_1) → G'
  16. Mirror(G, c_1) → G'_1
  17. Mirror(G, a_1) → G'_2
  18. LineBisector(G'_1, G') → h
  19. LineBisector(G'_2, G') → i
  20. Circle(A, C', B') → k
  21. Circle(B, C', A') → p
  22. Circle(C, A', B') → q
  23. Center(k) → D
  24. Center(p) → E
  25. Center(q) → F
  26. Polygon(E, D, F) → poly2, f_1, e_1, d_1
  27. Midpoint(d_1) → I
  28. Midpoint(e_1) → J
  29. Segment(D, I) → j
  30. Segment(E, J) → l
  31. Midpoint(f_1) → K
  32. Segment(K, F) → m
  33. Intersect(d, g, 2) → S

AI Drawing Prompt Reference

Draw a triangle ABC and its circumcircle d. Find the centroid G of triangle ABC, and construct the orthogonal lines e and f through points A and B, respectively, perpendicular to sides BC and AC, intersecting at point H, which is the Steiner point. Then, draw the circumcenter O of the circumcircle d, and construct the image triangle A'B'C' of triangle ABC with respect to G. Draw the circles g that are the images of A'B'C' with respect to each side. Construct the images of the Steiner point H with respect to each side, G', G'_1, and G'_2, and form a new triangle E'D'F' with these images. Find the centroids I, J, and K of triangle E'D'F', and draw the segments j, l, and m connecting these centroids. Finally, find the second intersection point S of the circumcircle d and the image circle g.
The prompt above was compiled from the original drawing commands and is for reference only.