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Centroid, Orthocenter, and Circumcenter of a Triangle by Dylan

Centroid, Orthocenter, and Circumcenter of a Triangle by Dylan
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2015-11-04 07:40
TriangleCirclePolygonGeometric Construction
This GeoGebra example demonstrates the centroid, orthocenter, and circumcenter of a triangle and their geometric relationships. It covers core concepts such as medians, altitudes, perpendicular bisectors, triangle centers, and the circumcircle. The construction uses Polygon to create triangle poly1, Point to define vertices A, B, C and intersection points, Line to draw medians, altitudes, or perpendicular bisectors, Circle to construct the circumcircle, and Intersect to locate centers and key points. Suitable for middle-school geometry, high-school Euclidean geometry, and exploratory math labs.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Point(poly1) → D
  3. Circle(A, Segment[A, D]) → d
  4. Circle(C, Segment[A, D]) → e
  5. Intersect(d, e, 2) → E
  6. Intersect(d, e, 1) → F
  7. Line(E, F) → f
  8. Point(poly1) → G
  9. Circle(B, Segment[B, G]) → g
  10. Circle(A, Segment[B, G]) → h
  11. Intersect(g, h, 2) → H
  12. Intersect(g, h, 1) → I
  13. Line(H, I) → i
  14. Point(poly1) → J
  15. Circle(B, Segment[B, J]) → k
  16. Circle(C, Segment[B, J]) → p
  17. Intersect(k, p, 1) → K
  18. Intersect(k, p, 2) → L
  19. Line(K, L) → j
  20. Intersect(f, i) → M
  21. Circle(B, N) → q
  22. Intersect(q, b, 1) → O
  23. Intersect(q, b, 2) → P
  24. Circle(O, D) → r
  25. Circle(P, Segment[O, D]) → s
  26. Intersect(r, s, 1) → Q
  27. Intersect(r, s, 2) → R
  28. Line(Q, R) → l
  29. Circle(C, S) → t
  30. Intersect(t, c, 1) → T
  31. Circle(B, U) → c_1
  32. Circle(T, Segment[B, U]) → d_1
  33. Intersect(c_1, d_1, 2) → V
  34. Line(V, C) → m
  35. Circle(A, W) → e_1
  36. Intersect(e_1, a, 2) → Z
  37. Intersect(e_1, a, 1) → A_1
  38. Point(e_1) → B_1
  39. Circle(Z, B_1) → f_1
  40. Circle(A_1, Segment[Z, B_1]) → g_1
  41. Intersect(f_1, g_1, 2) → C_1
  42. Line(C_1, A) → n
  43. Intersect(f, b) → D_1
  44. Intersect(j, a) → E_1
  45. Intersect(i, c) → F_1
  46. Line(F_1, C) → a_1
  47. Line(D_1, B) → b_1
  48. Line(E_1, A) → h_1
  49. Intersect(a_1, b_1) → G_1
  50. Line(M, G_1) → i_1
  51. Intersect(t, c, 2) → H_1
  52. Intersect(l, m) → I_1

AI Drawing Prompt Reference

Draw a triangle with three vertices. Pick a point inside the triangle, draw a circle centered at that point with radius to one vertex, and another circle centered at another vertex with the same radius; the line through their two intersections is constructed. Repeat this with two other interior points to get three such lines. These lines intersect pairwise at three points on the triangle's sides. Then, from one vertex, draw a circle with a suitable radius intersecting the opposite side, and from that intersection draw another circle with the same radius; the line through their intersections is a perpendicular. Construct two more perpendiculars similarly. Finally, each perpendicular meets its corresponding side at a foot, and lines from each vertex to the opposite foot form the three altitudes.
The prompt above was compiled from the original drawing commands and is for reference only.