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Circumcenter and Circumcircle Dynamic Construction

Circumcenter and Circumcircle Dynamic Construction
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2020-11-20 23:59
TriangleCircleAngleTransformationGeometric Construction
This GeoGebra applet dynamically demonstrates the construction of the circumcenter and circumcircle of a triangle. The core mathematical concept is the circumcenter: the point equidistant from all three vertices, which is the intersection of the perpendicular bisectors of the triangle's sidescars. The construction uses commands such as Polygon to create triangle ABC, Midpoint and OrthogonalLine to draw perpendicular bisectors, Intersect to find the circumcenter D, and Circle or CircleArc to draw the circumcircle centered at D with radius DA. Additional commands like Rotate, Angle, Ray, Segment, and Dilate are used to illustrate related angles and dynamic variations. This applet is suitable for middle and high school geometry classes, helping students visualize the relationship between a triangle and its circumcircle, and explore how the circumcenter moves as the triangle changes shape.

GeoGebra Commands & Parameters

  1. Rotate(B, (Angle * π / 180), A) → B'
  2. Angle(B, A, B') → α
  3. Ray(A, B') → f
  4. Segment(A, B) → g
  5. Point(f) → C
  6. Polygon(B, A, C) → t1, c, b, a_1
  7. Segment(A, C) → h
  8. Segment(C, B) → i
  9. Midpoint(g) → D
  10. Midpoint(i) → E
  11. Midpoint(h) → F
  12. OrthogonalLine(E, i) → j
  13. OrthogonalLine(D, g) → k
  14. OrthogonalLine(F, h) → l
  15. Intersect(j, k) → G
  16. Dilate(A, If[0 < d ≤ 1, 1 - d, 0], D) → A'
  17. Segment(A, A') → m
  18. Rotate(A, If[1 < d ≤ 2, ((-(d - 1)) * π), (-π)], D) → A'_1
  19. Segment(D, A'_1) → n
  20. Dilate(B, If[2 < d ≤ 3, 3 - d, 0], E) → D'
  21. Segment(B, D') → p
  22. Rotate(B, If[3 < d ≤ 4, ((-(d - 3)) * π), (-π)], E) → B'_1
  23. Segment(E, B'_1) → q
  24. Dilate(C, If[4 < d ≤ 5, 5 - d, 0], F) → C'
  25. Segment(C, C') → r
  26. Rotate(C, If[5 < d ≤ 6, ((-(d - 5)) * π), (-π)], F) → C'_1
  27. Segment(F, C'_1) → s
  28. Dilate(D, If[6 < d ≤ 7, 7 - d, 0], G) → D'_1
  29. Segment(D, D'_1) → t
  30. Angle(G, D, A) → β
  31. Dilate(E, If[7 < d ≤ 8, 8 - d, 0], G) → E'
  32. Segment(E, E') → e
  33. Point(j) → H
  34. Angle(H, E, B) → γ
  35. Dilate(F, If[8 < d ≤ 9, 9 - d, 0], G) → F'
  36. Segment(F, F') → f_1
  37. Angle(G, F, C) → δ
  38. Dilate(G, If[10 < d ≤ 11, 11 - d, 0], C) → G'
  39. Segment(G, G') → g_1
  40. Angle(C, G, A) → ε
  41. Angle(A, G, B) → ζ
  42. Rotate(G, If[11 < d ≤ 12, ((d - 11) * ε), ε], G) → G'_1
  43. Rotate(G', If[11 < d ≤ 12, ((d - 11) * ε), ε], G) → G''
  44. Segment(G'_1, G'') → h_1
  45. CircleArc(G, C, G'') → c_1
  46. Rotate(G'_1, If[12 < d ≤ 13, ((d - 12) * ζ), ζ], G) → I
  47. Rotate(G'', If[12 < d ≤ 13, ((d - 12) * ζ), ζ], G) → G'''
  48. Segment(I, G''') → i_1
  49. CircleArc(G, A, G''') → d_1
  50. Rotate(I, If[13 < d ≤ 14, ((d - 13) * ((2 * π) - ε - ζ)), (2 * π) - ε - ζ], G) → I'
  51. Rotate(G''', If[13 < d ≤ 14, ((d - 13) * ((2 * π) - ε - ζ)), (2 * π) - ε - ζ], G) → J
  52. Segment(I', J) → j_1
  53. CircleArc(G, B, J) → e_1

AI Drawing Prompt Reference

Draw a triangle and dynamically demonstrate its circumcircle and circumcenter. First, sketch the triangle's three sides, then find the midpoints of each side and draw the perpendicular bisectors. The intersection of these bisectors is the circumcenter. Draw a circle centered at the circumcenter, passing through any vertex of the triangle to show the circumcircle. Additionally, demonstrate how the circumcenter moves and the relationships between angles by rotating and scaling the triangle's vertices dynamically.
The prompt above was compiled from the original drawing commands and is for reference only.