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Geometric Mean Altitude Theorem

Geometric Mean Altitude Theorem
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2019-01-05 02:47
TriangleTrigonometry
This example illustrates the Geometric Mean Altitude Theorem, which states that in a right triangle, the altitude to the hypotenuse is the geometric mean of the two segments of the hypotenuse. Key mathematical concepts include geometric mean and triangle properties. Using GeoGebra commands such as Point, Line, OrthogonalLine, and Intersect, the construction of a right triangle, its altitude, and the projection lines is demonstrated. Suitable for middle school and above students to learn geometric construction and related theorems.

GeoGebra Commands & Parameters

  1. Point(f) → A
  2. Line(A, B) → g
  3. OrthogonalLine(B, g) → h_1
  4. Intersect(f, h_1) → C
  5. OrthogonalLine(B, f) → i
  6. Intersect(f, i) → D
  7. Segment(A, C) → j
  8. Segment(C, B) → k
  9. Segment(B, A) → l
  10. Segment(D, B) → h
  11. Angle(C, D, B) → α
  12. Angle(A, B, C) → β
  13. Segment(A, D) → a
  14. Segment(D, C) → b
  15. Polygon(A, B, D) → t1, d, a_1, b_1
  16. Dilate(D, If[0 ≤ c < 1, 1 - c, 0], E) → E'
  17. Vector(D, E') → v
  18. Translate(A, v) → A'
  19. Translate(B, v) → B'
  20. Translate(D, v) → D'
  21. Polygon(A', B', D') → t1', d', a', b'
  22. Rotate(A', If[1 ≤ c < 2, ((1 - c) * π / 2), (3 * π / 2)], E) → A''
  23. Rotate(B', If[1 ≤ c < 2, ((1 - c) * π / 2), (3 * π / 2)], E) → B''
  24. Rotate(D', If[1 ≤ c < 2, ((1 - c) * π / 2), (3 * π / 2)], E) → D''
  25. Polygon(A'', B'', D'') → t1'', d'', a'', b''
  26. Polygon(D, C, B) → t2, b_2, d_1, c_1
  27. Dilate(D, If[2 ≤ c < 3, 3 - c, 0], F) → F'
  28. Vector(D, F') → u
  29. Translate(D, u) → D'_1
  30. Translate(C, u) → C'
  31. Translate(B, u) → B'_1
  32. Polygon(D'_1, C', B'_1) → t2', b'_1, d'_1, c'
  33. Angle(B', D', A') → γ
  34. Angle(B'', D'', A'') → δ
  35. Angle(C', D'_1, B'_1) → ε

AI Drawing Prompt Reference

Draw a right triangle ABC with AB as the hypotenuse and CD as the altitude from C to AB. Construct a perpendicular line h1 through B, and find the intersection C of h1 with AB. Then, construct a perpendicular line i through B to BC, and find the intersection D of i with AB, which is the foot of the altitude CD. Connect AC, BC, and BD to form the right triangle ABC and the altitude CD.
The prompt above was compiled from the original drawing commands and is for reference only.