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Proof of the Law of Sines by Albert Navetta

Proof of the Law of Sines by Albert Navetta
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2019-08-18 23:10
TriangleTrigonometryGeometric Construction
This example demonstrates the proof of the Law of Sines, which relates the lengths of the sides of a triangle to the sines of its angles. By constructing the altitudes of the triangle and using the properties of similar triangles, the proof is derived. The example uses commands such as Polygon, PointIn, Angle, OrthogonalLine, Line, Intersect, Segment, and Corner in GeoGebra, suitable for high school mathematics education.

GeoGebra Commands & Parameters

  1. Polygon((-9.08831568900743, 1.4996534836992197), (110.91168431099257, 1.4996534836992197), (110.91168431099257, 81.49965348369926), (-9.08831568900743, 81.49965348369926)) → tv, j, k, l, m
  2. PointIn(tv) → B
  3. PointIn(tv) → C
  4. PointIn(tv) → A
  5. Polygon(A, B, C) → poly1, c, a, b
  6. Angle(C, B, A) → angB
  7. Angle(A, C, B) → angC
  8. Angle(B, A, C) → angA
  9. OrthogonalLine(B, b) → d
  10. Line(A, C) → e
  11. Intersect(d, e) → Base1
  12. Segment(Base1, B) → h1
  13. Segment(C, Base1) → ext1
  14. Segment(A, Base1) → ext2
  15. Angle(B, Base1, C) → h1ang1
  16. Angle(A, Base1, B) → h1ang2
  17. Line(A, B) → f
  18. OrthogonalLine(C, f) → g
  19. Intersect(f, g) → Base2
  20. Segment(B, Base2) → ext3
  21. Segment(C, Base2) → h2
  22. Angle(C, Base2, A) → h2ang1
  23. Segment(A, Base2) → ext4
  24. Angle(B, Base2, C) → h2ang2
  25. Polygon(Base1, B, C) → trih11, c_1, base1, b_1
  26. Polygon(Base1, A, B) → trih12, b_2, h, a_1
  27. Polygon(A, Base2, C) → trih21, c_2, a_2, base2
  28. Polygon(Base2, B, C) → trih22, c_3, i, b_3
  29. Corner(5) → Thedims

AI Drawing Prompt Reference

Draw a triangle and erect perpendiculars from each vertex to the opposite sides, forming three right-angled triangles. Measure the lengths of the opposite sides and hypotenuses of these triangles, and label the angles of the main triangle. Use the properties of similar triangles to demonstrate the proportional relationship between the sides of the large triangle and the sines of its corresponding angles, proving the Law of Sines.
The prompt above was compiled from the original drawing commands and is for reference only.