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Midpoint of Hypotenuse Demonstration

Midpoint of Hypotenuse Demonstration
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2016-05-11 05:41
TriangleCircleGeometric Construction
This applet dynamically illustrates the property that the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices without measuring distances or angles. It involves core concepts like right triangles, hypotenuse midpoints, and equidistance. Constructed using commands such as Line, OrthogonalLine, Point, Ray, Midpoint, Segment, Circle, Intersect, CircleSector, Polygon, Rotate, Dilate, and Angle. Suitable for high school mathematics education to visually understand the midpoint theorem of the hypotenuse.

GeoGebra Commands & Parameters

  1. Line(A, B) → a
  2. OrthogonalLine(A, a) → d
  3. OrthogonalLine(B, a) → e
  4. Point(d) → C
  5. Ray(A, C) → c
  6. Point(d) → E
  7. Ray(A, E) → b
  8. Ray(A, B) → g
  9. Point(g) → F
  10. Ray(A, F) → h
  11. Point(h) → G
  12. Ray(A, E) → i
  13. Point(i) → H
  14. Midpoint(H, G) → D
  15. Segment(A, H) → j
  16. Segment(A, G) → k
  17. Circle(A, Min[(0.15 * j), (0.15 * k)]) → p
  18. Intersect(p, a, 2) → I
  19. Intersect(p, i, 1) → J
  20. Intersect(p, a, 1) → K
  21. CircleSector(A, J, K) → q
  22. Polygon(A, G, H) → poly1, h_1, a_1, g_1
  23. Rotate(q, If[0 < f ≤ 1, ((-f) * π / 2), -1.5707963267948966], A) → q'
  24. Dilate(G, If[1 < f ≤ 2, 2 - f, 0], D) → G'
  25. Segment(G, G') → l
  26. Rotate(G, If[2 < f ≤ 3, ((f - 2) * π), 3.141592653589793], D) → G'_1
  27. Rotate(G', If[2 < f ≤ 3, ((f - 2) * π), 3.141592653589793], D) → G''
  28. Segment(G'_1, G'') → l'
  29. Angle(H, D, A) → α
  30. Rotate(G'_1, If[3 < f ≤ 4, ((f - 3) * α), α], D) → L
  31. Rotate(G'', If[3 < f ≤ 4, ((f - 3) * α), α], D) → G'''
  32. Segment(L, G''') → l''

AI Drawing Prompt Reference

Draw a right triangle with one leg horizontal and the other vertical. Pick a point on the vertical leg as the third vertex, and connect it to the two endpoints of the horizontal leg to form the triangle. Find the midpoint of the hypotenuse, and draw a circle centered at this midpoint passing through all three vertices, showing that the midpoint is equidistant from them. Add some auxiliary arcs and segments for a dynamic demonstration, without labeling specific angles or lengths.
The prompt above was compiled from the original drawing commands and is for reference only.