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Trigonometric Equations Related to Proposition 13 of Aristarchus' On Sizes

Trigonometric Equations Related to Proposition 13 of Aristarchus' On Sizes
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2020-11-11 02:41
CircleAngleTrigonometrySolid GeometryGeometric Construction
This GeoGebra example demonstrates a geometric construction related to trigonometric equations derived from Proposition 13 of Aristarchus' ancient work 'On Sizes'. It involves concepts from spherical geometry, similar triangles, trigonometric functions, and properties of intersections between lines and circles. The construction uses commands such as Point, Rotate, Angle, Circle, CircumcircleArc, and Segment, creating numerous objects including points (B, S, Z, M, V, A, O, N, L, C, J, G, U, D), segments (BN, BC, ZA, ZB), arcs, and circles. Rotations and angle measurements are employed to establish the trigonometric relationships. This resource is suitable for high school or university students studying trigonometry, geometric construction, and the history of mathematics. It can be used to verify ancient propositions through modern dynamic geometry or as a case study in mathematical modeling and interactive geometry instruction.

GeoGebra Commands & Parameters

  1. Point(yAxis) → B
  2. Rotate(Z, φ''', B) → Z'
  3. Angle(Z, B, Z') → φ''
  4. Circle(B, e) → c
  5. Circle(A, s) → d'
  6. Circle(A, s) → d''
  7. Angle(A, B, V) → ρ'''
  8. Circle(B, M) → mo
  9. Circle(B, O) → r
  10. CircumcircleArc(O, L, N) → OLN
  11. Circle(C, L) → t
  12. Angle(C, B, L) → ρ''
  13. Segment(B, N) → BN
  14. Segment(B, C) → BC
  15. Angle(G, B, J) → ζ''
  16. Segment(L, C) → m
  17. Segment(O, N) → ON
  18. Segment(B, U) → BU
  19. Segment(B, V) → BV
  20. Segment(Z, D) → ZD
  21. Segment(Z, F) → ZF
  22. Angle(D, Z, A) → ζ'
  23. Angle(U, A, D) → γ'
  24. Segment(Z, O) → ZO
  25. Segment(O, B) → OB
  26. Segment(Z, B) → ZB''
  27. Angle(B, O, Z) → ω
  28. Segment(L, R) → LR'
  29. Segment(L, N) → LN''
  30. Segment(U, V) → UV'
  31. Segment(Z, C) → ZC'
  32. Segment(Z, R) → ZR'
  33. Circle(H, E) → h
  34. Segment(B, E) → g
  35. Angle(E, B, H) → δ'
  36. Circle(P, I) → h'
  37. Segment(P, I) → f
  38. Segment(H, E) → i
  39. Circle(K, i) → k
  40. Circle(K, f) → c_1
  41. Angle(L, B, O) → φ
  42. Circle(B, BC') → d_1
  43. Segment(B, O'') → n

AI Drawing Prompt Reference

Draw a geometric figure related to Proposition 13 of Aristarchus' 'On Sizes'. Start by creating point B on the y-axis, then rotate point Z around B to get Z', and measure angle ZBZ' to get φ'''. Next, draw circle c with B as the center, and draw circles d' and d'' with A as the center, and measure angle ABV to get ρ'''. Draw circle mo and r, construct arc OLN, draw circle t, and measure angle CBL to get ρ''. Draw segments BN, BC, ZD, ZF, BU, BV, ZC, ZR, ZD, ZF, ZB, OB, ZB', OLN, t, d', d'', c, h, h', k, c_1, φ, d_1, and n, and measure angle ZBO to get ω, and draw segments LR', LN'', UV', ZC', ZR', g, i, f, k, c_1, δ', φ'', φ, d_1, and n.
The prompt above was compiled from the original drawing commands and is for reference only.