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If the lengths of the perpendiculars from the vertices of a triangle ABC on the opposite sides are `p_(1), p_(2), p_(3)` then prove that `(1)/(p_(1)) + (1)/(p_(2)) + (1)/(p_(3)) = (1)/(r) = (1)/(r_(1)) + (1)/(r_(2)) + (1)/(r_(3))`.

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To prove the statement \( \frac{1}{p_1} + \frac{1}{p_2} + \frac{1}{p_3} = \frac{1}{r} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \), we will break down the proof into clear steps. ### Step-by-Step Solution: **Step 1: Express \( \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} \)** We know that the inradius \( r \) of triangle \( ABC \) can be expressed in terms of the semi-perimeter \( s \) and the area \( \Delta \) of the triangle: \[ ...
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