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Circumcenter and Circumcircle Construction Quiz

Circumcenter and Circumcircle Construction Quiz
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2020-05-06 18:32
TriangleCircleAngleCoordinate GeometryGeometric Construction
① This GeoGebra applet dynamically demonstrates the construction of the circumcircle of a triangle, followed by key questions as an interactive quiz. ② The core mathematical concept is the circumcenter—the intersection point of the perpendicular bisectors of a triangle's sides, equidistant from all three vertices, which uniquely defines the circumcircle. ③ Construction details: from the summary, commands such as Rotate, Angle, Ray, Segment, Point, Polygon, Midpoint, OrthogonalLine, Intersect, Dilate, and CircleArc are used. Objects include triangle vertices A, B, C; side midpoints D, E, F; perpendicular lines h, i, j; intersection point G (the circumcenter); and the circumcircle d. The applet illustrates the classic steps: construct perpendicular bisectors of two sides, find their intersection, then draw the circle centered at that point with radius to any vertex. ④ Suitable for middle school geometry lessons or self-study, helping students grasp the properties of the circumcenter, the uniqueness of the circumcircle, and the integration of dynamic construction with verification. It also serves as a ready-made quiz for classroom assessment.

GeoGebra Commands & Parameters

  1. Rotate(B, (Unghi * 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