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Animation 269: Dynamic Geometry of Circles and Secants with Calculus Connections

Animation 269: Dynamic Geometry of Circles and Secants with Calculus Connections
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2017-05-24 22:04
CircleCalculusDerivativeIntegralSlider Animation
This GeoGebra example constructs a dynamic geometric scene involving circles, secant lines, arcs, points, midpoints, rotations, and dilations, demonstrating how geometric relationships evolve as elements move. It integrates concepts from calculus (derivatives, integrals), percentages, and ratios, making it a rich visual tool for exploring the interplay between geometry and analysis. The construction uses commands such as Circle, Segment, PointIn, Distance, Line, Intersect, CircleArc, Point, Midpoint, Dilate, and Rotate, creating numerous objects (A, B, c, f, a, C, b, g, D, E, d, e, h, k, F, G, p, H, i, I, J, K, q, j, L, M, l, A', m, K', n, A'_1, A'', m', K'_1, r, K'_2, N, r', s) to track changes in lengths, angles, areas, or ratios as a slider or animation drives the motion. Suitable for high school and undergraduate students, this example helps visualize secant properties, dynamic loci, and the fundamental ideas of derivatives and integrals through geometric transformations. It also serves as a model for creating interactive math lessons.

GeoGebra Commands & Parameters

  1. Circle(A, B) → c
  2. Segment(A, B) → f
  3. PointIn(a) → C
  4. Distance(C, c) → b
  5. Line(A, C) → g
  6. Intersect(c, g, 2) → D
  7. Intersect(c, g, 1) → E
  8. Distance(C, D) → d
  9. Distance(C, E) → e
  10. Circle(C, h) → k
  11. Intersect(k, c) → F, G, ,
  12. CircleArc(A, G, F) → p
  13. Point(p) → H
  14. Line(H, G) → i
  15. Intersect(k, i) → I, J
  16. Midpoint(A, C) → K
  17. Circle(K, G) → q
  18. Line(H, G) → j
  19. Intersect(q, j) → L, M
  20. Dilate(A, If[0 < l ≤ 1, 1 - l, 0], K) → A'
  21. Segment(A, A') → m
  22. Dilate(K, If[1 < l ≤ 2, 2 - l, 0], C) → K'
  23. Segment(K, K') → n
  24. Rotate(A, If[2 < l ≤ 3, ((l - 2) * π), π], K) → A'_1
  25. Rotate(A', If[2 < l ≤ 3, ((l - 2) * π), π], K) → A''
  26. Segment(A'_1, A'') → m'
  27. Dilate(K, If[3 < l ≤ 4, 4 - l, 0], G) → K'_1
  28. Segment(K, K'_1) → r
  29. Rotate(K, If[4 < l ≤ 6, ((l - 4) * π), (2 * π)], K) → K'_2
  30. Rotate(K'_1, If[4 < l ≤ 6, ((l - 4) * π), (2 * π)], K) → N
  31. Segment(K'_2, N) → r'
  32. CircleArc(K, G, N) → s
  33. Dilate(H, If[6 < l ≤ 7, 7 - l, 0], M) → H'_1
  34. Segment(H, H'_1) → t
  35. Dilate(M, If[7 < l ≤ 8, 8 - l, 0], J) → M'
  36. Segment(M, M') → f_1
  37. Rotate(H, If[8 < l ≤ 9, ((l - 8) * π), π], M) → H'
  38. Rotate(H'_1, If[8 < l ≤ 9, ((l - 8) * π), π], M) → O
  39. Segment(H', O) → t'