सिद्ध कीजिए कि - `(a^2sin(B-C))/(sinB+sinC)+(b^2sin(C-A))/(sinC+sinA)+(c^2sin(A-B))/(sinA+sinB)=0`

Explanation

हम जानते है कि `(sinA)/(a)=(sinB)/(b)=(sinC)/(c)=k`
`rArr sin A=ak , sin B =bk, sin C=ck `
`therefore (a^2sin(B-C))/(sinB+sinC)=(a^2(sinB cos C-cosB sinC))/(bk+ck)`
`=(a^2)/(k(b+c))[bk((a^2+b^2-c^2)/(2ab))-ck ((c^2+a^2-b^2)/(2ca))]`
`=(a^2k)/(2ak(b+c))[(a^2+b^2-c^2)-(c^2+a^2-b^2)]`
`=(a)/(2ak(b+c))[a^2+b^2-c^2-c^2-a^2+b^2]`
`=(a)/(2(b+c))[2(b^2-c^2)]`
`=(a)/((b+c))(b+c)(b-c)=a(b-c)` ...(i)
इसी प्रकार `(b^2sin(C-A))/(sinC+sinA)=b(c-a)`.....(ii)
तथा `(c^2sin(A-B))/(sinA+sinB)=c(a-b)` ....(iii)
समीकरण (i),(ii)व (iii) को जोड़ने पर `(a^2sin(B-C))/(sinB+sinC)+(b^2sin(C-A))/(sinC+sinA)+(c^2sin(A-B))/(sinA+sinB)=a(b-c)(c-a)+c(a-b)=0` यही सिद्ध करना था ।