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Explore Properties of Tangents to a Circle

Explore Properties of Tangents to a Circle
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2019-06-29 05:29
CircleAngleGeometric Construction
This applet helps students discover the properties of tangents drawn to a circle from a point outside the circle. Key mathematical concepts include the properties of tangents to a circle. The construction uses GeoGebra commands such as Circle, Tangent, Intersect, etc. Suitable for students in middle school and above for geometric construction and circle properties.

GeoGebra Commands & Parameters

  1. Circle(O, B_1) → c
  2. Segment(O, B_1) → a
  3. PointIn(b) → E
  4. Tangent(E, c) → d, e
  5. Intersect(c, e, 1) → B
  6. Intersect(c, d, 1) → A
  7. Ray(E, A) → f
  8. Ray(E, B) → g
  9. Segment(O, A) → h
  10. Segment(O, B) → i
  11. Dilate(O, If[0 < j ≤ 1, 1 - j, 0], A) → A'
  12. Dilate(O, If[0 < j ≤ 1, 1 - j, 0], B) → A'_1
  13. Segment(O, A') → k
  14. Segment(O, A'_1) → l
  15. Angle(E, B, O) → α
  16. Angle(O, A, E) → β
  17. Circle(A, (0.25 * a)) → p
  18. Circle(B, (0.25 * a)) → q
  19. Intersect(p, h, 1) → F
  20. Intersect(q, i, 1) → G
  21. Intersect(p, f) → H, I
  22. Intersect(q, g) → J, K
  23. Circle(O, B_1) → t
  24. Dilate(O, If[1.3 < j ≤ 2.3, 2.3 - j, 0], E) → C'
  25. Polygon(B, E, O) → poly1, a_1, d_1, c_1
  26. Angle(A, E, B) → γ
  27. Rotate(B, If[5 < j ≤ 6, ((j - 5) * ((2 * π) - γ)), (2 * π) - γ], E) → D'_1
  28. Rotate(E, If[5 < j ≤ 6, ((j - 5) * ((2 * π) - γ)), (2 * π) - γ], E) → C'_1
  29. Segment(E, A) → n
  30. Segment(E, B) → b_1
  31. Polygon(O, E, A) → poly2, e_1, a_2, c_2
  32. Segment(O, C') → f_1
  33. Rotate(E, If[5 < j ≤ 6, ((j - 5) * (-γ)), (-γ)], E) → C'_2
  34. Rotate(B, If[2.5 < j ≤ 3.5, ((j - 2.5) * (-γ)), (-γ)], E) → D'
  35. Segment(C'_2, D') → h_1

AI Drawing Prompt Reference

Draw a circle with center O and a point B₁ on it, connecting O to B₁ as a radius. Place a point E outside the circle, and draw two tangents from E to the circle, touching at points A and B. Connect E to A and B, and also connect O to A and B. On segments OA and OB, place points A' and B' such that their distances from O are scaled by a factor. Display the angles AEB, EBO, and OAE. Draw small circles centered at A and B, intersecting the segments and tangents, and mark the intersection points. Construct triangles OBE and OEA, and include a rotating angle to dynamically explore the tangent properties.
The prompt above was compiled from the original drawing commands and is for reference only.