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Square with Concyclic Points and Angle Bisector

Square with Concyclic Points and Angle Bisector
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2019-07-27 20:46
CirclePolygonAngle
This example illustrates a problem posted by Antonio Gutierrez involving a square, concyclic points, and angle bisectors. Using GeoGebra commands such as rotation, ray, and intersection, it dynamically shows the relationship between the angle bisector and concyclic points. Suitable for students at the junior high school level and above, focusing on geometric transformations and circle properties.

GeoGebra Commands & Parameters

  1. Polygon(A, B, D, C) → q1, a_1, b, d, c
  2. Segment(A, B) → f
  3. Segment(B, D) → g
  4. Segment(D, C) → h
  5. Point(Ray[(0, 0, 0), (1, 0, 0)]) → E
  6. Segment(C, A) → i
  7. Segment(C, E) → j
  8. Angle(E, C, D, xOyPlane) → α
  9. Rotate(E, α / 2, C) → E'
  10. Angle(E, C, E', xOyPlane) → β
  11. Ray(C, E') → k
  12. Intersect(g, k) → F
  13. Ray(C, E) → l
  14. OrthogonalLine(F, l, xOyPlane) → m
  15. Intersect(m, l) → G
  16. Angle(F, G, C, xOyPlane) → γ
  17. Ray(A, G) → n
  18. Intersect(k, n) → H
  19. Center(Circle[A, B, C]) → I
  20. Circle(I, D) → e
  21. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], E) → C'
  22. Segment(C, C') → p
  23. Dilate(C, If[1 < o ≤ 2, 2 - o, 0], F) → C'_1
  24. Ray(C, C') → q
  25. Segment(C, C'_1) → r
  26. Ray(C, H) → s
  27. Circle(C, (0.3 * f)) → t
  28. Intersect(t, h) → J,
  29. Rotate(J, If[1 < o ≤ 2, ((-(o - 1)) * β), (-β)], C) → J'
  30. CircleSector(C, J', J) → c_1
  31. Rotate(c_1, If[3 < o ≤ 4, ((-(o - 3)) * β), (-β)], C) → d_1
  32. Dilate(F, If[5 < o ≤ 6, 6 - o, 0], G) → F'
  33. Segment(F, F') → f_1
  34. Rotate(D, If[6 < o ≤ 8, ((o - 6) * π), (2 * π)], I) → D'
  35. CircleArc(I, D, D') → e_1
  36. Dilate(A, If[8.5 < o ≤ 9.5, 9.5 - o, 0], G) → A'
  37. Segment(A, A') → g_1
  38. Dilate(G, If[9.5 < o ≤ 10.5, 10.5 - o, 0], H) → G'
  39. Segment(G, G') → h_1
  40. Circle(C, D, A) → k_1
  41. Ray(A, B) → i_1

AI Drawing Prompt Reference

Draw a square and add an angle bisector inside it that intersects one side at a point. From this point, draw a ray and find its intersection with the other side of the square. With one vertex of the square as the center, draw a circle passing through the other two vertices. Finally, find the point where the ray intersects the circle and connect these points to show the relationship between the angle bisector and the circle.
The prompt above was compiled from the original drawing commands and is for reference only.