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Another Parabola Theorem

Another Parabola Theorem
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2016-10-17 05:34
TrigonometryGeometric Construction
In collections Conic Sections
This example demonstrates an interesting theorem about a parabola. It involves the symmetry of parabolas, isosceles triangles, and trigonometric functions. By constructing elements such as parabolas, vertices, points, and tangents, one can gain a better understanding of these concepts. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Line(A, B) → d
  2. Parabola(F, d) → c
  3. Vertex(c) → V, , ,
  4. Line(V, d) → g
  5. Point(c) → P
  6. OrthogonalLine(P, g) → h
  7. Intersect(g, h) → F_1
  8. Tangent(P, c) → i,
  9. Intersect(g, i) → G
  10. Intersect(d, h) → C
  11. Circle(P, G) → d_1
  12. Intersect(d_1, i, 2) → D
  13. Segment(P, F_1) → j
  14. Circle(F_1, (0.2 * j)) → e
  15. Intersect(e, g, 2) → H
  16. Intersect(e, j, 1) → I
  17. Segment(F_1, C) → l
  18. Circle(C, (0.2 * j)) → p
  19. Intersect(p, d, 2) → J
  20. Intersect(p, l, 1) → K
  21. Intersect(p, d, 1) → L
  22. CircleSector(C, K, L) → q
  23. Dilate(P, If[0 < a ≤ 1, 1 - a, 0], C) → E'
  24. Segment(P, E') → k
  25. Rotate(q, If[1.5 < a ≤ 2.5, ((-(a - 1.5)) * π / 2), ((-π)) / 2], C) → q'
  26. Dilate(C, If[3 < a ≤ 4, 4 - a, 0], F_1) → C'
  27. Vector(C, C') → u
  28. Translate(q', u) → q''
  29. Dilate(P, If[4 < a ≤ 5, 5 - a, 0], G) → E'_1
  30. Segment(P, E'_1) → t
  31. Dilate(V, If[5 < a ≤ 6, 6 - a, 0], G) → V'
  32. Segment(V, V') → m
  33. Rotate(V, If[6 < a ≤ 7, ((a - 6) * π), π], G) → V'_1
  34. Rotate(V', If[6 < a ≤ 7, ((a - 6) * π), π], G) → V''
  35. Segment(V'_1, V'') → m'
  36. CircleSector(C, L, K) → r
  37. Rotate(r, If[1.5 < a ≤ 2.5, ((a - 1.5) * π / 2), π / 2], C) → r'
  38. Translate(r', u) → r''