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(A) Since the area of a triangle is given by $\Delta = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{altitude}$,we have: $\Delta = \frac{1}{2} a \alpha

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$\sin^2 A + \sin^2 B + \sin^2 C$

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(A) Since the area of a triangle is given by $\Delta = \frac{1}{2} 	imes 	ext{base} 	imes 	ext{altitude}$,we have: $\Delta = \frac{1}{2} a \alpha = \frac{1}{2} b \beta = \frac{1}{2} c \gamma$ $\Rightarrow \alpha = \frac{2 \Delta}{a}, \beta = \frac{2 \Delta}{b}, \gamma = \frac{2 \Delta}{c}$ Now,consider the expression: $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}ight)$ $= \frac{\Delta^2}{R^2}\left(\frac{a^2}{4 \Delta^2} + \frac{b^2}{4 \Delta^2} + \frac{c^2}{4 \Delta^2}ight)$ $= \frac{\Delta^2}{R^2} \cdot \frac{1}{4 \Delta^2} (a^2 + b^2 + c^2)$ $= \frac{1}{4 R^2} (a^2 + b^2 + c^2)$ Using the sine rule,$a = 2R \sin A, b = 2R \sin B, c = 2R \sin C$: $= \frac{1}{4 R^2} ((2R \sin A)^2 + (2R \sin B)^2 + (2R \sin C)^2)$ $= \frac{1}{4 R^2} (4R^2 \sin^2 A + 4R^2 \sin^2 B + 4R^2 \sin^2 C)$ $= \sin^2 A + \sin^2 B + \sin^2 C$

If $\alpha, \beta$ and $\gamma$ are the lengths of the altitudes of a $\triangle ABC$ with area $\Delta$,then $\frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$ is equal to

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