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Proof of Sine Law

Proof of Sine Law
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2014-10-10 09:21
TriangleAngleFunctionTrigonometryGeometric Construction
This example dynamically demonstrates the proof of the Sine Law. It illustrates that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Key math concepts include trigonometric functions, triangle geometry, and geometric construction. The example uses GeoGebra commands such as Segment, OrthogonalLine, Polygon, Distance, Line, Point, Intersect, and Angle to construct the figure and showcase the validity of the theorem. Suitable for high school mathematics students to learn and understand the Sine Law and its applications.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a_2
  2. Segment(B, C) → a
  3. Segment(C, A) → b
  4. OrthogonalLine(C, a_2) → d
  5. Polygon(A, B, C) → poly1, c_1, a_1, b_1
  6. Distance(A, C) → distanceAC
  7. Distance(B, C) → distanceBC
  8. Distance(A, B) → distanceAB
  9. Line(A, B) → e
  10. OrthogonalLine(C, e) → f
  11. Point(e) → D
  12. Intersect(f, e) → E
  13. Segment(C, E) → h
  14. Angle(D, E, C) → α
  15. Distance(E, C) → distanceEC
  16. Segment(A, C) → h_1
  17. Segment(C, B) → i

AI Drawing Prompt Reference

Draw a triangle and show the perpendicular from a vertex to the opposite side, marking the foot of the perpendicular and the corresponding angle. Measure the lengths of the sides and the sine values of the opposite angles, verifying that the ratio of each side length to its sine is constant. Demonstrate dynamically that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant, illustrating the Sine Law.
The prompt above was compiled from the original drawing commands and is for reference only.