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

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

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We know that by sine rule: `a/sinA​,b/sinB,c/sinC​=k`
L.H.S=`(a^2sin(B-C))/(sinB+ sin C)+(b^2sin(C-A))/(sinC+ sin A)+(c^2sin(A-B))/(sinA+ sin B)`
=`(k^2 sin^2A sin(B-C))/(sinB+ sin C)+(k^2sin^2 B sin(C-A))/(sinC+ sin A)+(k^2 sin^2Csin(A-B))/(sinA+ sin B)`
=`(k^2 sinA [sinAsin(B-C)])/(sinB+ sin C)+(k^2sin B[sinB sin(C-A)])/(sinC+ sin A)+(k^2 sinC[ sinCsin(A-B)])/(sinA+ sin B)`
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