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Minimal Square Containing Triangle PQR

Minimal Square Containing Triangle PQR
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2014-05-23 07:22
TriangleVectorCoordinate GeometryOptimization
This example demonstrates how to find the optimal/minimal square containing a given triangle PQR through interactive manipulation. It involves core mathematical concepts like vectors, rotation, coordinate geometry, and optimization. Key GeoGebra commands and objects used include Polygon, Sort, Element, CircumcircleSector, Point, Vector, Function, Intersect, RemoveUndefined, Sequence, IndexOf, Max. Suitable for high school and above students to understand geometric transformations and optimization problems.

GeoGebra Commands & Parameters

  1. Polygon(P, Q, R) → triangle, r, p, q
  2. Sort({Length[p], Length[q], Length[r]}) → sides
  3. Element(sides, 3) → a
  4. Element(sides, 2) → b
  5. Element(sides, 1) → c
  6. CircumcircleSector((a, 0), ((a * sqrt(2) / 2), (a * sqrt(2) / 2)), (0, a)) → d
  7. Point(d) → rotate
  8. Vector(rotate) → u
  9. Vector(endpoint) → v
  10. Polygon((0, 0), rotate, endpoint) → copy, b_1, e, a_1
  11. Function((a * cos(x)), 0, 1.5707963267948966) → Δx_1
  12. Function((a * cos(x)) - (b * cos(x + θ)), 0, 1.5707963267948966) → Δx_2
  13. Function((a * sin(x)), 0, 1.5707963267948966) → Δy_1
  14. Function((b * sin(x + θ)), 0, 1.5707963267948966) → Δy_2
  15. Intersect(Δx_1, Δx_2, 0, 1.5707963267948966) → A
  16. Intersect(Δx_1, Δy_2, 0, 1.5707963267948966) → C
  17. Intersect(Δx_2, Δy_1, 0, 1.5707963267948966) → D
  18. Intersect(Δx_2, Δy_2, 0, 1.5707963267948966) → E, G
  19. Intersect(Δy_1, Δy_2, 0, 1.5707963267948966) → F
  20. RemoveUndefined({x(A), x(B), x(C), x(D), x(E), x(F)}) → ts
  21. Sequence(Max[{Δx_1(Element[ts, i]), Δx_2(Element[ts, i]), Δy_1(Element[ts, i]), Δy_2(Element[ts, i])}], i, 1, Length[ts]) → squares
  22. IndexOf(Min[squares], squares) → indeks
  23. Element(ts, indeks) → t_{optimal}
  24. Polygon((0, 0), ((a * cos(t_{optimal})), (a * sin(t_{optimal}))), ((b * cos(t_{optimal} + θ)), (b * sin(t_{optimal} + θ)))) → polygon1, g, h, i
  25. Element(squares, indeks) → optimalside
  26. Polygon(O, O + ((-optimalside), 0), O + ((-optimalside), optimalside), O + (0, optimalside)) → optimalsquare, o, j, k, l
  27. Max({Δx_1(t), Δx_2(t), Δy_1(t), Δy_2(t)}) → currentside
  28. Polygon(M, M + ((-currentside), 0), M + ((-currentside), currentside), M + (0, currentside)) → currentsquare, m, f, n, s

AI Drawing Prompt Reference

Draw a square that contains a triangle defined by three points. First, sketch the triangle, then find the minimal square enclosing it. Determine the position and size of the square by calculating the triangle's sides and rotation. Show the relationship between the square and the triangle, and how to optimize to find the optimal solution.
The prompt above was compiled from the original drawing commands and is for reference only.