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In any triangle A B C , prove that: (a^2...

In any triangle `A B C ,` prove that: `(a^2sin(B-C))/(sinB+s in C)+(b^2sin(C-A))/(sinC+s in A)+(c^2sin(A-B))/(sinA+s in B)=0`

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`(a^(2) sin (B - C))/(sin B + sin C) = (4R^(2) sin ^(2) A sin (B - C))/(sin B + sin C) ( :' a = 2R sin A)`
`= (4R^(2) sin A sin (B + C) sin (B - C))/(sin B + sin C)`
`= (4R^(2) sin A (sin^(2) B - sin^(2) C))/(sin B + sin C)`
`= 4R^(2) sin A (sin B - sin C)`
(i) similarly, `(b^(2) sin (C - A))/(sin C + sin A) = 4R^(2) sin B (sin C - sin A)`
and `(c^(2) sin (A - B))/(sin A + sin B) = 4R^(2) sin C (sin A - sin B)`
Adding we get
`(a^(2) sin (B - C))/(sin B + sin C) + (b^(2) sin (C - A))/(sin C + sin A) + (c^(2) sin (A - B))/(sin A + sin B) = 0`
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