If the altitudes of a triangle are in arithme | vedclass

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(B) In $	riangle ABC$,let $a, b, c$ be the side lengths and $p_1, p_2, p_3$ be the corresponding altitudes.We know that the area $\Delta = \frac{1}{2

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$HP$

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(B) In $	riangle ABC$,let $a, b, c$ be the side lengths and $p_1, p_2, p_3$ be the corresponding altitudes.We know that the area $\Delta = \frac{1}{2} a p_1 = \frac{1}{2} b p_2 = \frac{1}{2} c p_3$.This implies $p_1 = \frac{2\Delta}{a}$,$p_2 = \frac{2\Delta}{b}$,and $p_3 = \frac{2\Delta}{c}$.Given that $p_1, p_2, p_3$ are in $AP$,we have $\frac{2\Delta}{a}, \frac{2\Delta}{b}, \frac{2\Delta}{c}$ in $AP$.Dividing by $2\Delta$,we get $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $AP$.By definition,if the reciprocals of terms are in $AP$,then the terms themselves are in $HP$.Therefore,$a, b, c$ are in $HP$.

If the altitudes of a triangle are in arithmetic progression,then the sides of the triangle are in

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