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If $\alpha, \beta$ and $\gamma$ are length of the altitudes of a $\triangle A B C$ 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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$\sin ^2 A+\sin ^2 B+\sin ^2 C$
$\because$ Area of triangle $=\frac{1}{2}$ base $\times$ altitude
$\therefore \Delta=\frac{1}{2} a \alpha=\frac{1}{2} b \beta=\frac{1}{2} c \gamma$
$\Rightarrow \alpha=\frac{2 \Delta}{a}, \beta=\frac{2 \Delta}{b}$ and $\gamma=\frac{2 \Delta}{c}$
$\therefore \frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$
$=\frac{\Delta^2}{R^2}\left(\frac{a^2}{4 \Delta^2}+\frac{b^2}{4 \Delta^2}+\frac{c^2}{4 \Delta^2}\right)$
$\begin{aligned} & =\frac{1}{4 R^2}\left(4 R^2 \sin ^2 A+4 R^2 \sin ^2 B+4 R^2 \sin ^2 C\right) \\ & \qquad\left[\because \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2 R}\right] \\ & =\sin ^2 A+\sin ^2 B+\sin ^2 C\end{aligned}$
$\therefore \Delta=\frac{1}{2} a \alpha=\frac{1}{2} b \beta=\frac{1}{2} c \gamma$
$\Rightarrow \alpha=\frac{2 \Delta}{a}, \beta=\frac{2 \Delta}{b}$ and $\gamma=\frac{2 \Delta}{c}$
$\therefore \frac{\Delta^2}{R^2}\left(\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}\right)$
$=\frac{\Delta^2}{R^2}\left(\frac{a^2}{4 \Delta^2}+\frac{b^2}{4 \Delta^2}+\frac{c^2}{4 \Delta^2}\right)$
$\begin{aligned} & =\frac{1}{4 R^2}\left(4 R^2 \sin ^2 A+4 R^2 \sin ^2 B+4 R^2 \sin ^2 C\right) \\ & \qquad\left[\because \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2 R}\right] \\ & =\sin ^2 A+\sin ^2 B+\sin ^2 C\end{aligned}$
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