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Geometric Mean Flash in a Parabola

Geometric Mean Flash in a Parabola
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2018-10-18 10:32
FunctionCoordinate GeometryTransformation
This example demonstrates the flash of geometric mean within a parabola. It involves key concepts of geometric mean and parabolas. Using GeoGebra commands such as Point, Line, Intersect, Circle, Semicircle, Midpoint, OrthogonalLine, Dilate, Segment, Vector, Translate, Rotate, CircleArc, Angle, etc., this phenomenon can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  1. Point(g) → A
  2. Point(h) → B
  3. Line(A, B) → i
  4. Intersect(yAxis, i) → C
  5. Circle(B, y(A)) → c
  6. Intersect(c, j, 1) → D
  7. Intersect(j, xAxis) → E
  8. Semicircle(D, E) → d
  9. Midpoint(E, D) → F
  10. OrthogonalLine(B, j) → k
  11. Intersect(d, k, 1) → J
  12. Dilate(H, If[0 < a ≤ 1, 1 - a, 0], A) → H'
  13. Dilate(E, If[0 < a ≤ 1, 1 - a, 0], B) → E'
  14. Segment(H, H') → l
  15. Segment(E, E') → m
  16. Dilate(A, If[2 < a ≤ 3, 3 - a, 0], D) → A'
  17. Vector(A, A') → u
  18. Translate(H, u) → H'_1
  19. Translate(H', u) → H''
  20. Segment(H'_1, H'') → l'
  21. Rotate(E, If[3 < a ≤ 5, ((a - 3) * π / 2), π], F) → E'_1
  22. CircleArc(F, E, E'_1) → e
  23. Dilate(G, If[5 < a ≤ 6, 6 - a, 0], C) → G'
  24. Segment(G, G') → n
  25. Angle(E, B, J) → α
  26. Dilate(J, If[6 < a ≤ 7, 7 - a, 0], B) → J'
  27. Segment(J, J') → p
  28. Segment(A, B) → q
  29. Dilate(G, If[7.5 < a ≤ 8.5, 8.5 - a, 0], J) → G'_1
  30. Vector(G, G'_1) → v
  31. Translate(G, v) → G'_2
  32. Translate(G', v) → G''
  33. Segment(G'_2, G'') → n'
  34. Rotate(G'_2, If[7.5 < a ≤ 8.5, ((a - 7.5) * π / 2), π / 2], G'_2) → K
  35. Rotate(G'', If[7.5 < a ≤ 8.5, ((a - 7.5) * π / 2), π / 2], G'_2) → G'''
  36. Segment(K, G''') → n''

AI Drawing Prompt Reference

Draw a parabola and find two points A and B on it. Connect these two points with a line that intersects the y-axis at point C. With point B as the center and the y-coordinate of point A as the radius, draw a circle that intersects line j at point D. Line j intersects the x-axis at point E. With points D and E as the endpoints, draw a semicircle d. Find the midpoint F of semicircle d, and draw a line through point B that is perpendicular to line j, intersecting semicircle d at point J. According to the value of parameter a, scale point H and E with point A or B as the center to get points H' and E', then connect H and H', and E and E'. With point D as the center, scale point A to get point A', then draw a vector u through points A and A', and translate point H along vector u to get points H'_1 and H''. With point F as the center, rotate point E to get point E'_1, then draw an arc e with point F as the center and points E and E'_1 as endpoints. With point C as the center, scale point G to get point G', then connect G and G'. With point B as the center, scale point J to get point J', then connect J and J'. With point A as the center, scale point G to get point G'_1, then draw a vector v through points G and G'_1, and translate point G'_1 along vector v to get points G'_2 and G'', then connect G'_2 and G''. With point G'_2 as the center, rotate point G'_2 to get point K, and rotate point G'' to get point G''', then connect K and G'''.
The prompt above was compiled from the original drawing commands and is for reference only.