If p_1, p_2, p_3 are altitudes of a triangle | vedclass

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(C) We know that the area of a triangle $\Delta = \frac{1}{2} a p_1 = \frac{1}{2} b p_2 = \frac{1}{2} c p_3$. From this,we get $p_1 = \frac{2\Delta}{a

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$\frac{s - c}{\Delta}$

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(C) We know that the area of a triangle $\Delta = \frac{1}{2} a p_1 = \frac{1}{2} b p_2 = \frac{1}{2} c p_3$. From this,we get $p_1 = \frac{2\Delta}{a}$,$p_2 = \frac{2\Delta}{b}$,and $p_3 = \frac{2\Delta}{c}$. Therefore,$p_1^{-1} + p_2^{-1} - p_3^{-1} = \frac{1}{p_1} + \frac{1}{p_2} - \frac{1}{p_3} = \frac{a}{2\Delta} + \frac{b}{2\Delta} - \frac{c}{2\Delta}$. This simplifies to $\frac{a + b - c}{2\Delta}$. Since the semi-perimeter $s = \frac{a + b + c}{2}$,we have $a + b + c = 2s$,so $a + b - c = (a + b + c) - 2c = 2s - 2c = 2(s - c)$. Substituting this into the expression,we get $\frac{2(s - c)}{2\Delta} = \frac{s - c}{\Delta}$.

If $p_1, p_2, p_3$ are altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively,and $\Delta$ is the area of the triangle,then $p_1^{-1} + p_2^{-1} - p_3^{-1}$ is equal to

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