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Properties of Tangents

Properties of Tangents
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2019-12-02 14:57
CircleAngleGeometric Construction
This example demonstrates the properties of two tangents drawn from a point outside a circle. It involves core mathematical concepts like tangent lines, circles, angles, and similar triangles. Construction highlights include using GeoGebra commands such as Circle, Segment, PointIn, Tangent, Intersect, Ray, Dilate, Angle, Polygon, and Rotate. Suitable for high school mathematics instruction to explore the tangent length theorem and the property that tangents are perpendicular to radii at the point of contact.

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