In a Delta ABC,the expression left( frac{a^2} | vedclass

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(B) Using the Sine Rule,we have $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.Thus,the expression becomes $(a(2R) + b(2R) + c(2R)) \si

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$\Delta$

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(B) Using the Sine Rule,we have $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.Thus,the expression becomes $(a(2R) + b(2R) + c(2R)) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$.$= 2R(a + b + c) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$.Since $a+b+c = 2s$,we have $2R(2s) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} = 4Rs \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$.We know that $\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} = \frac{r}{4R}$,where $r$ is the inradius.Substituting this,we get $4Rs \left( \frac{r}{4R} \right) = rs$.Since $\Delta = rs$,the expression simplifies to $\Delta$.

In a $\Delta ABC$,the expression $\left( \frac{a^2}{\sin A} + \frac{b^2}{\sin B} + \frac{c^2}{\sin C} \right) \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2}$ simplifies to,where $\Delta$ is the area of the triangle.

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