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Pythagoras' Theorem Part 3

Pythagoras' Theorem Part 3
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2015-05-17 17:23
TriangleCircleVectorGeometric ConstructionSlider Animation
In collections Euclid's proof of Pythagoras Theorem (proposition 47) Pythagoras' theorem
This example demonstrates the application of Pythagoras' theorem in three-dimensional space by constructing right triangles and related vectors to showcase the geometric significance of the theorem. Core mathematical concepts include vectors, triangles, hypergeometric distribution, rational numbers, and congruence. The construction utilizes commands such as Segment, OrthogonalLine, Point, Angle, Circle, Intersect, Polygon, and PointIn in GeoGebra. It is suitable for high school students and educators to explore the integration of spatial geometry and algebraic relationships.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. OrthogonalLine(A, a) → b
  3. Point(b) → C
  4. Segment(C, A) → c
  5. Segment(C, B) → d
  6. Angle(B, A, C) → α
  7. Circle(B, A) → e
  8. Circle(A, B) → f
  9. OrthogonalLine(B, a) → g
  10. OrthogonalLine(A, a) → h
  11. Intersect(f, h, 1) → D
  12. Intersect(e, g, 1) → E
  13. Segment(D, E) → i
  14. Segment(B, A) → j
  15. Segment(A, D) → k
  16. Segment(E, B) → l
  17. Circle(A, C) → p
  18. Circle(C, A) → q
  19. OrthogonalLine(A, c) → m
  20. Intersect(p, m, 1) → F
  21. OrthogonalLine(C, c) → n
  22. Intersect(q, n, 1) → G
  23. OrthogonalLine(C, d) → r
  24. OrthogonalLine(B, d) → s
  25. Circle(B, C) → t
  26. Circle(C, B) → c_1
  27. Intersect(c_1, r, 2) → H
  28. Intersect(t, s, 2) → I
  29. Segment(B, I) → a_1
  30. Segment(I, H) → b_1
  31. Segment(H, C) → d_1
  32. Segment(C, G) → e_1
  33. Segment(F, A) → f_1
  34. Segment(F, G) → g_1
  35. OrthogonalLine(A, d) → h_1
  36. Intersect(h_1, d) → J
  37. Intersect(h_1, b_1) → K
  38. Segment(A, J) → i_1
  39. Segment(J, K) → j_1
  40. Polygon(J, B, I, K) → poly1, j_2, b_2, i_2, k_1
  41. Polygon(I, A, B) → poly2, b_3, i_3, a_2
  42. PointIn(poly1) → Pointpoly1
  43. PointIn(poly2) → Pointpoly2

AI Drawing Prompt Reference

In three-dimensional space, draw a right triangle ABC where AB and AC are line segments, and BC is the hypotenuse. Construct a line perpendicular to AB through point A, intersecting at point C. Draw segments from C to A and from C to B, and mark angle alpha. Draw two circles with centers at B and A, with radii AB and AC. Find intersection points D and E of these circles, and connect DE. Draw segment from B to A, and segments from A to D and from B to E. Draw two circles with centers at A and C, with radii AC and BC. Find intersection points F and G of these circles. Draw segment from B to C, and segments from B to D and from C to E. Connect these points to form polygons, and check if points are inside polygons. Show the application of Pythagoras' theorem in three-dimensional space.
The prompt above was compiled from the original drawing commands and is for reference only.