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#56 Distorted Star of David with Parallelograms

#56 Distorted Star of David with Parallelograms
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2017-07-25 00:56
PolygonCoordinate GeometryTransformationGeometric Construction
① This example demonstrates a distorted Star of David composed of multiple parallelograms, showcasing a geometric construction with symmetry and transformations. ② Core mathematical concepts include polygons, parallelograms, central symmetry, reflection, translation, and point construction, involving triangles, quadrilaterals, and logical relationships. ③ Construction highlights: using GeoGebra commands such as Segment, Polygon, Centroid, Mirror, Intersect, PointIn, Line, Translate, and PointList, it constructs various points (e.g., C, B, A, D, M, N) and objects (poly1, poly3, POLY1), applying mirror, translation, and centroid operations to generate symmetric parallelogram arrangements that form the distorted star. ④ Suitable for middle/high school geometry lessons or math enrichment, helping students understand symmetry transformations, polygon combinations, and dynamic geometric construction, as well as exploring parallelogram properties and star-pattern generation logic.

GeoGebra Commands & Parameters

  1. Segment(A, C) → d
  2. Polygon(A, B, C) → poly1, c, a, b
  3. Centroid(poly1) → D
  4. Mirror(C, D) → C'
  5. Mirror(B, D) → B'
  6. Segment(C', B') → m
  7. Intersect(m, d) → M
  8. Mirror(A, D) → A'
  9. Segment(A', B') → l
  10. Intersect(l, d) → N
  11. Segment(A, A + (t1 * (C - A))) → f
  12. Segment(B', B' + (t3 * (A' - B'))) → i
  13. Intersect(f, i) → F
  14. Segment(C', C' + (t3 * (B' - C'))) → j
  15. Intersect(f, j) → E
  16. Segment(B, B + (t1 * (A - B))) → h
  17. Intersect(h, j) → J
  18. PointIn(Polygon[B', M, N]) → G
  19. Line(G, i) → q
  20. Intersect(q, j) → I
  21. Segment(C, C + (t1 * (B - C))) → g
  22. Line(G, j) → p
  23. Intersect(p, i) → H
  24. Polygon(B', I, G, H) → poly3, b', i_1, g_1, h_1
  25. Translate(B', Vector[B', B' + (t6 * (J - B'))]) → B''
  26. Translate(G, Vector[B', B' + (t6 * (J - B'))]) → G'
  27. Translate(H, Vector[B', B' + (t6 * (J - B'))]) → H'
  28. Translate(I, Vector[B', B' + (t6 * (J - B'))]) → I'
  29. Polygon(B'', I', G', H') → poly3', b'', i', g', h'
  30. Intersect(g, i) → K
  31. Translate(B', Vector[B', B' + (t6 * (K - B'))]) → B''_1
  32. Translate(I, Vector[B', B' + (t6 * (K - B'))]) → I'_1
  33. Translate(G, Vector[B', B' + (t6 * (K - B'))]) → G'_1
  34. Translate(H, Vector[B', B' + (t6 * (K - B'))]) → H'_1
  35. Polygon(B''_1, I'_1, G'_1, H'_1) → poly3'_1, b''_1, i'_1, g'_1, h'_1
  36. Segment(C, C + (t7 * (G' - C))) → r
  37. Segment(B, B + (t7 * (G - B))) → s
  38. Intersect(r, s) → L
  39. Segment(A, A + (t7 * (G'_1 - A))) → t
  40. Segment(A', A' + (t3 * (C' - A'))) → k
  41. PointList({B', M, N}) → list1
  42. PointIn(Polygon[B', M, N]) → O