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Desargues's Theorem Dynamic Illustration

Desargues's Theorem Dynamic Illustration
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2017-03-02 10:37
TriangleTransformationGeometric Construction
In collections CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Triangle Theorems
This example dynamically illustrates one conditional statement of Desargues's Theorem. By dragging points, you can observe that when the lines joining corresponding vertices of two triangles meet at a point, the lines joining the corresponding points of intersection of the sides are concurrent. Key mathematical concepts include triangles, collinear points, and perspective relationships. The construction uses commands such as Line, Point, Circle, Dilate, Segment, Intersect, and Polygon in GeoGebra. Suitable for high school geometry education.

GeoGebra Commands & Parameters

  1. Line(A_1, A') → f
  2. Line(A_1, C') → g
  3. Line(A_1, B') → h
  4. Point(g) → C
  5. Point(g) → C'_1
  6. Point(f) → A'_1
  7. Point(f) → A
  8. Point(h) → B
  9. Point(h) → B'_1
  10. Circle(A'_1, 1) → r
  11. Circle(A, 1) → t
  12. Dilate(A_1, If[0 < a ≤ 1, 1 - a, 0], A) → N
  13. Dilate(A_1, If[0 < a ≤ 1, 1 - a, 0], A') → O
  14. Segment(A_1, N) → b
  15. Segment(A_1, O) → i_1
  16. Dilate(A_1, If[1 < a ≤ 2, 2 - a, 0], B') → P
  17. Dilate(A_1, If[1 < a ≤ 2, 2 - a, 0], B) → Q
  18. Segment(A_1, P) → l_1
  19. Segment(A_1, Q) → m_1
  20. Dilate(A_1, If[2 < a ≤ 3, 3 - a, 0], C') → H
  21. Dilate(A_1, If[2 < a ≤ 3, 3 - a, 0], C) → I
  22. Segment(A_1, H) → c
  23. Segment(A_1, I) → d
  24. Line(A', B') → j
  25. Line(B, A) → k
  26. Intersect(j, k) → D
  27. Line(C', B') → l
  28. Line(B, C) → m
  29. Intersect(l, m) → E
  30. Line(C', A') → n
  31. Line(A, C) → p
  32. Intersect(n, p) → F
  33. Polygon(A, C, B) → poly1, b_1, a_1, c_1
  34. Polygon(A', B', C') → poly3, c', a', b'
  35. Dilate(B, If[5 < a ≤ 6, 6 - a, 0], D) → B'_2
  36. Segment(B, B'_2) → i
  37. Dilate(A', If[5 < a ≤ 6, 6 - a, 0], D) → A''
  38. Segment(A', A'') → q
  39. Dilate(B, If[6 < a ≤ 7, 7 - a, 0], E) → B'_3
  40. Dilate(C', If[6 < a ≤ 7, 7 - a, 0], E) → C''
  41. Segment(B, B'_3) → s
  42. Segment(C', C'') → e
  43. Dilate(A, If[7 < a ≤ 8, 8 - a, 0], F) → A'_2
  44. Dilate(C', If[7 < a ≤ 8, 8 - a, 0], F) → C''_1
  45. Segment(C', C''_1) → f_1
  46. Segment(A, A'_2) → g_1
  47. Dilate(D, If[10 < a ≤ 11, 11 - a, 0], E) → D'
  48. Dilate(E, If[11 < a ≤ 12, 12 - a, 0], F) → E'
  49. Segment(D, D') → h_1
  50. Segment(E, E') → j_1

AI Drawing Prompt Reference

Draw two triangles ABC and A'B'C', and connect their corresponding vertices with lines AA', BB', and CC' which intersect at point D. Through point D, draw lines parallel to the sides of triangle A'B'C' and observe if the lines intersecting with the sides of triangle ABC at points E and F also intersect at a point.
The prompt above was compiled from the original drawing commands and is for reference only.