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Smallest Square Enclosing a Triangle by Juan Vicente Sánchez

Smallest Square Enclosing a Triangle by Juan Vicente Sánchez
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2014-06-06 18:49
Coordinate GeometryOptimizationGeometric Construction
This example demonstrates the method for constructing the smallest square that encloses any given triangle. It involves core mathematical concepts such as coordinate geometry, geometric construction, and optimization. The construction utilizes GeoGebra commands and objects like Polygon, Sort, Intersect, Midpoint, Circle, LineBisector, If, Segment, OrthogonalLine, Ray, and Line. It is suitable for high school and above students to understand geometric optimization problems and construction techniques.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Sort({a, b, c}) → list1
  3. Intersect(Element[list1, 1], Element[list1, 2]) → D_1
  4. Intersect(Element[list1, 2], Element[list1, 3]) → E_1
  5. Intersect(Element[list1, 3], Element[list1, 1]) → F_1
  6. Midpoint(F_1, E_1) → G
  7. Circle(G, E_1) → d_1
  8. LineBisector(E_1, F_1) → e_2
  9. Intersect(d_1, e_2) → H, I
  10. Polygon(F_1, I, E_1, H) → poly2, f_1, i, e_1, h
  11. If(IsInRegion[D_1, poly2], H, D_1) → J
  12. Segment(E_1, F_1) → g_1
  13. Circle(J, g_1) → k
  14. OrthogonalLine(J, g_1) → j
  15. Intersect(j, g_1) → K
  16. Ray(J, K) → l
  17. Intersect(k, l) → L,
  18. Line(E_1, L) → m
  19. Intersect(d_1, m, 2) → D
  20. Line(F_1, D) → n
  21. OrthogonalLine(E_1, m) → p
  22. OrthogonalLine(J, n) → q
  23. Intersect(n, q) → E
  24. Intersect(p, q) → F
  25. Polygon(E_1, D, E, F) → poly3, d, e, f, g

AI Drawing Prompt Reference

Draw a triangle and then construct the smallest square that encloses it. Identify the midpoints of the triangle's sides, use them to create auxiliary circles and angle bisectors, and locate the vertices of the square. Connect these vertices to complete the square, ensuring it fully covers the triangle. Add necessary lines to illustrate the construction process and key geometric relationships.
The prompt above was compiled from the original drawing commands and is for reference only.