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Verifying the Law of Sines for Particular Cases: Step-by-Step Triangle Construct

Verifying the Law of Sines for Particular Cases: Step-by-Step Triangle Construct
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2015-05-22 11:35
TriangleCircleAngleTrigonometryGeometric Construction
This GeoGebra example guides students through a step-by-step construction of a triangle to verify the Law of Sines in specific cases. The core mathematical concept is the Law of Sines (a/sin A = b/sin B = c/sin C) and the relationships between sides and angles in a triangle. The construction uses commands such as Segment, Angle, Distance, Circle, Tangent, Intersect, and AngularBisector. Key objects include points A, B, C, sides a, b, c, angles α, β, γ, and multiple text objects (texto1 to texto11) that display side lengths and sine ratios. By dragging vertices, students can observe that the ratio of each side to the sine of its opposite angle remains constant, reinforcing the theorem. This resource is suitable for high school or introductory college courses in trigonometry or geometric construction.

GeoGebra Commands & Parameters

  1. Segment(A, B) → c
  2. Segment(B, C) → a
  3. Segment(C, A) → b
  4. Angle(A, B, C) → β
  5. Angle(B, C, A) → γ
  6. Angle(C, A, B) → α
  7. Distance(C, B) → distanciaCB
  8. Circle(B, C, A) → d
  9. Tangent(A, d) → e,
  10. Tangent(B, d) → f,
  11. Tangent(C, d) → g,
  12. Intersect(e, f) → D
  13. Intersect(g, f) → E
  14. Intersect(g, e) → F
  15. AngularBisector(D, E, F) → h
  16. AngularBisector(E, F, D) → i
  17. AngularBisector(F, D, E) → j
  18. Intersect(h, i) → O
  19. Distance(O, B) → distanciaGB
  20. Segment(B, O) → r
  21. Segment(O, C) → l
  22. Segment(O, A) → m
  23. Distance(O, C) → distanciaGC
  24. Distance(C, A) → distanciaCA
  25. Distance(O, B) → n

AI Drawing Prompt Reference

Draw a triangle with points A, B, C and sides a, b, c and angles α, β, γ. First, draw the triangle, then construct a circle and three tangents to find their intersection points. Next, construct angle bisectors from these points and find their intersection. Finally, add text objects to display side lengths and sine ratios. This figure is used to verify the Law of Sines.
The prompt above was compiled from the original drawing commands and is for reference only.