Prove by the method of vectors that in triangle

a/sin A = b/sin B = c/sin C

In Triangle ABC, The area of triangle ABC = 1/2*|ABxAC| = 1/2*|ABxBC| = 1/2*|BCxAC|

= 1/2*|AB||AC| sin A = = 1/2*|CB||AB| sin B= 1/2*|BC||AC| sin C

   b.c.sin(A)/2 = a.c.sin(B)/2 = a.b.sin(C)/2 
          b.c.sin(A) =a.c.sin(B) = a.b.sin(C)

dividing through by  a.b.c, we get  a/sin A = b/sin B = c/sin C
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Ramesh