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G_6.01 Quadrilaterals and other polygons

G_6.01 Quadrilaterals and other polygons
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2022-07-10 23:04
PolygonCoordinate GeometryGeometric Construction
This example demonstrates constructing various polygons, including quadrilaterals and other polygons using GeoGebra. Key mathematical concepts involve geometric construction, vectors, angles, and coordinate geometry. Construction steps include using commands like Sequence, Rotate, Polygon, Vector, Angle, and objects such as Point, Midpoint, Translate. Suitable for middle or high school mathematics education to understand polygon properties and geometric transformations.

GeoGebra Commands & Parameters

  1. Sequence(allVertices(k), k, 1, numOfSides) → polyVertices
  2. Sequence((x(allVertices(Mod[k, numOfSides] + 1)) + (Min[Max[t - (2 * k), 0], 1] * (x(allVertices(Mod[k + 1, numOfSides] + 1)) - x(allVertices(Mod[k, numOfSides] + 1)))), y(allVertices(Mod[k, numOfSides] + 1)) + (Min[Max[t - (2 * k), 0], 1] * (y(allVertices(Mod[k + 1, numOfSides] + 1)) - y(allVertices(Mod[k, numOfSides] + 1))))), k, 0, numOfSides) → Pnums
  3. Sequence(Rotate[(x(Pnums(k + 1)) + (x(allVertices(Mod[k + 1, numOfSides] + 1)) - x(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)], y(Pnums(k + 1)) + (y(allVertices(Mod[k + 1, numOfSides] + 1)) - y(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)]), (Min[Max[t - (2 * k) - 1, 0], 1] * (180° - Angle[allVertices(Mod[k + 2, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1), allVertices(Mod[k, numOfSides] + 1)])), Pnums(k + 1)], k, 0, numOfSides) → Pnums'
  4. Rotate((x(P6) + (x(A) - x(E)) / Distance[E, A], y(P6) + (y(A) - y(E)) / Distance[E, A]), (Min[Max[t - 11, 0], 1] * (180° - Angle[B, A, E])), P6) → P6'
  5. Rotate((x(P1) + (x(A) - x(E)) / Distance[E, A], y(P1) + (y(A) - y(E)) / Distance[E, A]), (Min[Max[t - 1, 0], 1] * (180° - Angle[B, A, E])), P1) → P1'
  6. Rotate((x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B]), (Min[Max[t - 3, 0], 1] * (180° - Angle[C, B, A])), P2) → P2'
  7. Rotate((x(P3) + (x(C) - x(B)) / Distance[B, C], y(P3) + (y(C) - y(B)) / Distance[B, C]), (Min[Max[t - 5, 0], 1] * (180° - Angle[D, C, B])), P3) → P3'
  8. Rotate((x(P4) + (x(D) - x(C)) / Distance[C, D], y(P4) + (y(D) - y(C)) / Distance[C, D]), (Min[Max[t - 7, 0], 1] * (180° - Angle[E, D, C])), P4) → P4'
  9. Rotate((x(P5) + (x(E) - x(D)) / Distance[E, D], y(P5) + (y(E) - y(D)) / Distance[E, D]), (Min[Max[t - 9, 0], 1] * (180° - Angle[A, E, D])), P5) → P5'
  10. Polygon(E, A, B, C, D) → poly1, a_1, b_1, c_1, d_1, e_1
  11. Vector(P, P') → u
  12. Angle(P1', P1, (x(P1) + (x(A) - x(E)) / Distance[E, A], y(P1) + (y(A) - y(E)) / Distance[E, A])) → α
  13. Angle(P2', P2, (x(P2) + (x(B) - x(A)) / Distance[A, B], y(P2) + (y(B) - y(A)) / Distance[A, B])) → β
  14. Angle(Pnums'(Mod[2, numOfSides] + 1), Pnums(Mod[2, numOfSides] + 1), (x(Pnums(Mod[2, numOfSides] + 1)) + (x(allVertices(Mod[3, numOfSides] + 1)) - x(allVertices(Mod[2, numOfSides] + 1))) / Distance[allVertices(Mod[2, numOfSides] + 1), allVertices(Mod[3, numOfSides] + 1)], y(Pnums(Mod[2, numOfSides] + 1)) + (y(allVertices(Mod[3, numOfSides] + 1)) - y(allVertices(Mod[2, numOfSides] + 1))) / Distance[allVertices(Mod[2, numOfSides] + 1), allVertices(Mod[3, numOfSides] + 1)])) → γ
  15. Angle(Pnums'(Mod[3, numOfSides] + 1), Pnums(Mod[3, numOfSides] + 1), (x(Pnums(Mod[3, numOfSides] + 1)) + (x(allVertices(Mod[4, numOfSides] + 1)) - x(allVertices(Mod[3, numOfSides] + 1))) / Distance[allVertices(Mod[3, numOfSides] + 1), allVertices(Mod[4, numOfSides] + 1)], y(Pnums(Mod[3, numOfSides] + 1)) + (y(allVertices(Mod[4, numOfSides] + 1)) - y(allVertices(Mod[3, numOfSides] + 1))) / Distance[allVertices(Mod[3, numOfSides] + 1), allVertices(Mod[4, numOfSides] + 1)])) → δ
  16. Angle(Pnums'(Mod[4, numOfSides] + 1), Pnums(Mod[4, numOfSides] + 1), (x(Pnums(Mod[4, numOfSides] + 1)) + (x(allVertices(Mod[5, numOfSides] + 1)) - x(allVertices(Mod[4, numOfSides] + 1))) / Distance[allVertices(Mod[4, numOfSides] + 1), allVertices(Mod[5, numOfSides] + 1)], y(Pnums(Mod[4, numOfSides] + 1)) + (y(allVertices(Mod[5, numOfSides] + 1)) - y(allVertices(Mod[4, numOfSides] + 1))) / Distance[allVertices(Mod[4, numOfSides] + 1), allVertices(Mod[5, numOfSides] + 1)])) → ε
  17. OrthogonalLine(P, u, xOyPlane) → f_{1}
  18. Point(f_{1}) → F
  19. Midpoint(F, P') → G
  20. Vector(G, P) → v
  21. Translate(P, v) → P'_1
  22. Translate(P', v) → P''
  23. Segment(A, (x(A) + (x(A) - x(E)) / Distance[E, A], y(A) + (y(A) - y(E)) / Distance[E, A])) → g
  24. Segment(B, (x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B])) → h
  25. Segment(C, (x(C) + (x(C) - x(B)) / Distance[B, C], y(C) + (y(C) - y(B)) / Distance[B, C])) → i
  26. Segment(D, (x(D) + (x(D) - x(C)) / Distance[C, D], y(D) + (y(D) - y(C)) / Distance[C, D])) → j
  27. Segment(E, (x(E) + (x(E) - x(D)) / Distance[D, E], y(E) + (y(E) - y(D)) / Distance[D, E])) → k
  28. Midpoint(a_1) → O
  29. Angle((x(A) + (x(A) - x(E)) / Distance[E, A], y(A) + (y(A) - y(E)) / Distance[E, A]), A, B) → a
  30. Angle((x(B) + (x(B) - x(A)) / Distance[A, B], y(B) + (y(B) - y(A)) / Distance[A, B]), B, C) → b
  31. Angle((x(allVertices(Mod[3, numOfSides] + 1)) + (x(allVertices(Mod[3, numOfSides] + 1)) - x(allVertices(Mod[2, numOfSides] + 1))) / Distance[allVertices(Mod[2, numOfSides] + 1), allVertices(Mod[3, numOfSides] + 1)], y(allVertices(Mod[3, numOfSides] + 1)) + (y(allVertices(Mod[3, numOfSides] + 1)) - y(allVertices(Mod[2, numOfSides] + 1))) / Distance[allVertices(Mod[2, numOfSides] + 1), allVertices(Mod[3, numOfSides] + 1)]), allVertices(Mod[3, numOfSides] + 1), allVertices(Mod[4, numOfSides] + 1)) → c
  32. Angle((x(allVertices(Mod[4, numOfSides] + 1)) + (x(allVertices(Mod[4, numOfSides] + 1)) - x(allVertices(Mod[3, numOfSides] + 1))) / Distance[allVertices(Mod[3, numOfSides] + 1), allVertices(Mod[4, numOfSides] + 1)], y(allVertices(Mod[4, numOfSides] + 1)) + (y(allVertices(Mod[4, numOfSides] + 1)) - y(allVertices(Mod[3, numOfSides] + 1))) / Distance[allVertices(Mod[3, numOfSides] + 1), allVertices(Mod[4, numOfSides] + 1)]), allVertices(Mod[4, numOfSides] + 1), allVertices(Mod[5, numOfSides] + 1)) → d
  33. Angle((x(allVertices(Mod[5, numOfSides] + 1)) + (x(allVertices(Mod[5, numOfSides] + 1)) - x(allVertices(Mod[4, numOfSides] + 1))) / Distance[allVertices(Mod[4, numOfSides] + 1), allVertices(Mod[5, numOfSides] + 1)], y(allVertices(Mod[5, numOfSides] + 1)) + (y(allVertices(Mod[5, numOfSides] + 1)) - y(allVertices(Mod[4, numOfSides] + 1))) / Distance[allVertices(Mod[4, numOfSides] + 1), allVertices(Mod[5, numOfSides] + 1)]), allVertices(Mod[5, numOfSides] + 1), allVertices(Mod[6, numOfSides] + 1)) → e
  34. If(t ≥ 1 ∧ t < 2, α, If[t ≥ 2, a, 0°]) → α'
  35. If(t ≥ 3 ∧ t < 4, β, If[t ≥ 4, b, 0°]) → β'
  36. If(t ≥ 5 ∧ t < 6, γ, If[t ≥ 6, c, 0°]) → γ'
  37. If(t ≥ 7 ∧ t < 8, δ, If[t ≥ 7, d, 0°]) → δ'
  38. If(t ≥ 9 ∧ t < 10, ε, If[t ≥ 9, e, 0°]) → ε'
  39. Polygon(polyVertices) → polySelect
  40. Sequence(Angle[(x(allVertices(Mod[k + 1, numOfSides] + 1)) + (x(allVertices(Mod[k + 1, numOfSides] + 1)) - x(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)], y(allVertices(Mod[k + 1, numOfSides] + 1)) + (y(allVertices(Mod[k + 1, numOfSides] + 1)) - y(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)]), allVertices(Mod[k + 1, numOfSides] + 1), allVertices(Mod[k + 2, numOfSides] + 1)], k, 0, floor(t / 2) - 1) → angleList
  41. Angle((x(allVertices(Mod[6, numOfSides] + 1)) + (x(allVertices(Mod[6, numOfSides] + 1)) - x(allVertices(Mod[5, numOfSides] + 1))) / Distance[allVertices(Mod[5, numOfSides] + 1), allVertices(Mod[6, numOfSides] + 1)], y(allVertices(Mod[6, numOfSides] + 1)) + (y(allVertices(Mod[6, numOfSides] + 1)) - y(allVertices(Mod[5, numOfSides] + 1))) / Distance[allVertices(Mod[5, numOfSides] + 1), allVertices(Mod[6, numOfSides] + 1)]), allVertices(Mod[6, numOfSides] + 1), allVertices(Mod[7, numOfSides] + 1)) → f
  42. Angle((x(allVertices(Mod[7, numOfSides] + 1)) + (x(allVertices(Mod[7, numOfSides] + 1)) - x(allVertices(Mod[6, numOfSides] + 1))) / Distance[allVertices(Mod[6, numOfSides] + 1), allVertices(Mod[7, numOfSides] + 1)], y(allVertices(Mod[7, numOfSides] + 1)) + (y(allVertices(Mod[7, numOfSides] + 1)) - y(allVertices(Mod[6, numOfSides] + 1))) / Distance[allVertices(Mod[6, numOfSides] + 1), allVertices(Mod[7, numOfSides] + 1)]), allVertices(Mod[7, numOfSides] + 1), allVertices(Mod[8, numOfSides] + 1)) → l
  43. Angle((x(allVertices(Mod[8, numOfSides] + 1)) + (x(allVertices(Mod[8, numOfSides] + 1)) - x(allVertices(Mod[7, numOfSides] + 1))) / Distance[allVertices(Mod[7, numOfSides] + 1), allVertices(Mod[8, numOfSides] + 1)], y(allVertices(Mod[8, numOfSides] + 1)) + (y(allVertices(Mod[8, numOfSides] + 1)) - y(allVertices(Mod[7, numOfSides] + 1))) / Distance[allVertices(Mod[7, numOfSides] + 1), allVertices(Mod[8, numOfSides] + 1)]), allVertices(Mod[8, numOfSides] + 1), allVertices(Mod[9, numOfSides] + 1)) → q
  44. Sequence(Segment[allVertices(Mod[k + 1, numOfSides] + 1), (x(allVertices(Mod[k + 1, numOfSides] + 1)) + (x(allVertices(Mod[k + 1, numOfSides] + 1)) - x(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)], y(allVertices(Mod[k + 1, numOfSides] + 1)) + (y(allVertices(Mod[k + 1, numOfSides] + 1)) - y(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)])], k, 0, floor((t + 1) / 2) - 1) → l1
  45. Angle(Pnums'(Mod[5, numOfSides] + 1), Pnums(Mod[5, numOfSides] + 1), (x(Pnums(Mod[5, numOfSides] + 1)) + (x(allVertices(Mod[6, numOfSides] + 1)) - x(allVertices(Mod[5, numOfSides] + 1))) / Distance[allVertices(Mod[5, numOfSides] + 1), allVertices(Mod[6, numOfSides] + 1)], y(Pnums(Mod[5, numOfSides] + 1)) + (y(allVertices(Mod[6, numOfSides] + 1)) - y(allVertices(Mod[5, numOfSides] + 1))) / Distance[allVertices(Mod[5, numOfSides] + 1), allVertices(Mod[6, numOfSides] + 1)])) → a6
  46. Angle(Pnums'(Mod[6, numOfSides] + 1), Pnums(Mod[6, numOfSides] + 1), (x(Pnums(Mod[6, numOfSides] + 1)) + (x(allVertices(Mod[7, numOfSides] + 1)) - x(allVertices(Mod[6, numOfSides] + 1))) / Distance[allVertices(Mod[6, numOfSides] + 1), allVertices(Mod[7, numOfSides] + 1)], y(Pnums(Mod[6, numOfSides] + 1)) + (y(allVertices(Mod[7, numOfSides] + 1)) - y(allVertices(Mod[6, numOfSides] + 1))) / Distance[allVertices(Mod[6, numOfSides] + 1), allVertices(Mod[7, numOfSides] + 1)])) → a7
  47. Angle(Pnums'(Mod[7, numOfSides] + 1), Pnums(Mod[7, numOfSides] + 1), (x(Pnums(Mod[7, numOfSides] + 1)) + (x(allVertices(Mod[8, numOfSides] + 1)) - x(allVertices(Mod[7, numOfSides] + 1))) / Distance[allVertices(Mod[7, numOfSides] + 1), allVertices(Mod[8, numOfSides] + 1)], y(Pnums(Mod[7, numOfSides] + 1)) + (y(allVertices(Mod[8, numOfSides] + 1)) - y(allVertices(Mod[7, numOfSides] + 1))) / Distance[allVertices(Mod[7, numOfSides] + 1), allVertices(Mod[8, numOfSides] + 1)])) → a8
  48. Sequence(Angle[Pnums'(Mod[k, numOfSides] + 1), Pnums(Mod[k, numOfSides] + 1), (x(Pnums(Mod[k, numOfSides] + 1)) + (x(allVertices(Mod[k + 1, numOfSides] + 1)) - x(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)], y(Pnums(Mod[k, numOfSides] + 1)) + (y(allVertices(Mod[k + 1, numOfSides] + 1)) - y(allVertices(Mod[k, numOfSides] + 1))) / Distance[allVertices(Mod[k, numOfSides] + 1), allVertices(Mod[k + 1, numOfSides] + 1)])], k, 0, numOfSides - 1) → eAngles
  49. Sum(eAngles) → theSum
  50. Corner(text16, 1) → t4c1
  51. Corner(text16, 2) → t4c2
  52. Corner(text17, 1) → t5c1
  53. Corner(text17, 2) → t5c2
  54. Corner(text18, 1) → t6c1
  55. Corner(text18, 2) → t6c2
  56. Corner(text16, 3) → t4c3
  57. Corner(text17, 3) → t5c3
  58. Corner(text18, 3) → t6c3
  59. Corner(text19, 1) → t7c1
  60. Corner(text19, 2) → t7c2
  61. Corner(text19, 3) → t7c3
  62. Corner(text21, 1) → t8c1
  63. Corner(text21, 2) → t8c2
  64. Corner(text21, 3) → t8c3

AI Drawing Prompt Reference

Draw a pentagon ABCDE by connecting points A to E, B to A, C to B, D to C, and E to D. Then construct vectors and midpoints, rotate points P1, P2, P3, P4, and P5, and finally draw the incircle and circumcircle of pentagon ABCDE.
The prompt above was compiled from the original drawing commands and is for reference only.