1.

In ` A B C ,`show that `a^2(s-a)+b^2(s-b)+c^(2)(s-c))=4R (a+r sin (A/2)sin( B/2)sin( C/2))`

Answer» `L.H.S. = a^2(s-a)+b^2(s-b)+c^2(s-c)`
Putting `s = (a+b+c)/2` in the above, we get,
`=1/2a^2(b+c-a)+1/2b^2(c+a-b)+1/2c^2(a+b-c)`
`=1/2(a(b^2+c^2-a^2)+b(c^2+a^2-b^2)+c(a^2+b^2-c^2))`
`=1/2(2abc cosA +2abc cosB+2abc cosC)`
`=abc (cosA+cosB+cos C)`
`=abc(1+4sin(A/2)sin(B/2)sin(C/2))`
As `Delta = (abc)/(4R), => abc = 4RDelta`
So, it becomes,
`= 4RDelta(1+4sin(A/2)sin(B/2)sin(C/2)) = R.H.S.`


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