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Dynamic Geometric Mean in Two Tangent Circles and a Common External Tangent (GoG

Dynamic Geometric Mean in Two Tangent Circles and a Common External Tangent (GoG
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2017-08-13 08:02
CircleAngleTransformationGeometric ConstructionSlider Animation
This interactive GeoGebra example dynamically demonstrates a geometric mean relationship among two mutually tangent circles, a common external tangent, and another special segment. The core mathematical concept is the geometric mean, where the length of a certain segment equals the square root of the product of two other segment lengths, intertwined with properties of circle tangents, similar triangles, and proportional relationships. The construction uses a rich set of GeoGebra commands including Circle, Point, Segment, Line, Intersect, Ray, Tangent, Semicircle, OrthogonalLine, Midpoint, Dilate, Angle, Rotate, CircleArc, Vector, and Translate, to build the two tangent circles, the external tangent, semicircles, perpendiculars, midpoints, rotations, and translations. Users can drag points or adjust sliders to explore how the geometric mean relation changes dynamically. This resource is ideal for high school or college geometry lessons focusing on geometric mean, tangent properties, and dynamic geometric constructions, and also serves as a valuable tool for mathematical experimentation and visualization.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Point(c) → C
  3. Segment(A, C) → f
  4. Line(A, C) → g
  5. Circle(C, 2) → d
  6. Intersect(d, g, 2) → E
  7. Ray(C, E) → h
  8. Point(h) → F
  9. Circle(F, C) → e
  10. Tangent(c, e) → i, j, k, l
  11. Intersect(-2.3471111943866765x² - 0.7116767516501961x y + 0.9621700797027014y² + 23.938748022851275x - 0.2856770767977977y = 52.59818359165294, c) → , , ,
  12. Intersect(c, i) → D
  13. Intersect(e, i) → G
  14. Semicircle(A, F) → p
  15. Line(A, F) → m
  16. OrthogonalLine(C, m) → n
  17. Intersect(p, n, 1) → H
  18. Segment(C, H) → q
  19. Segment(D, G) → r
  20. Midpoint(D, G) → I
  21. Dilate(A, If[2 < o ≤ 3, 3 - o, 0], D) → A'
  22. Dilate(F, If[2 < o ≤ 3, 3 - o, 0], G) → F'
  23. Segment(A, A') → s
  24. Segment(F, F') → t
  25. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], I) → D'
  26. Dilate(I, If[1 < o ≤ 2, 2 - o, 0], G) → I'
  27. Segment(D, D') → f_1
  28. Segment(I, I') → g_1
  29. Angle(I, D, A) → α
  30. Angle(F, G, I) → β
  31. Dilate(A, If[3 < o ≤ 4, 4 - o, 0], C) → A'_1
  32. Dilate(C, If[4 < o ≤ 5, 5 - o, 0], F) → C'
  33. Segment(A, A'_1) → h_1
  34. Segment(C, C') → i_1
  35. Rotate(F, If[5 < o ≤ 6, ((o - 5) * π), π], Midpoint[A, F]) → F'_1
  36. Midpoint(A, F) → J
  37. CircleArc(J, F, F'_1) → c_1
  38. Dilate(H, If[7 < o ≤ 8, 8 - o, 0], C) → H'
  39. Segment(H, H') → j_1
  40. Angle(H, C, A) → γ
  41. Segment(F, G) → k_1
  42. Angle(H, I, D) → δ
  43. Rotate(H, If[9 < o ≤ 10, ((o - 9) * δ), δ], C) → H'_1
  44. Rotate(H', If[9 < o ≤ 10, ((o - 9) * δ), δ], C) → H''
  45. Segment(H'_1, H'') → l_1
  46. Dilate(C, If[10 < o ≤ 11, 11 - o, 0], I) → C'_1
  47. Vector(C, C'_1) → u
  48. Translate(H'_1, u) → K
  49. Translate(H'', u) → H'''
  50. Segment(K, H''') → m_1
  51. Rotate(K, If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], I) → K'
  52. Rotate(H''', If[11 < o ≤ 12, ((-(o - 11)) * π), (-π)], I) → L
  53. Segment(K', L) → n_1

AI Drawing Prompt Reference

Draw two tangent circles with centers A and C, where the radius of the first circle is r and the radius of the second circle is 2r. Draw the external tangent line and the line segment AC. Construct a circle with center C and radius 2r, intersecting the external tangent at point E, and draw a ray EF from E. Construct a circle with center F and radius 2r, tangent to the first circle at points D and G. Draw a semicircle with center A and radius AF, and draw a perpendicular line from C to AF, intersecting the semicircle at point H. Draw segment CH and find its midpoint I. Dilate point A and F with D as the center and a given ratio to get A' and F', and draw segments AA' and FF'. Dilate point D with I as the center and a given ratio to get D', and draw segment DD'. Rotate point F to get F'1, and construct an arc c1 with D as the center and F'1 as the endpoint. Draw a circle with center C and radius 2r, intersecting the external tangent at point H', and draw segment CH', and find its midpoint K.
The prompt above was compiled from the original drawing commands and is for reference only.