To find the value of \( \frac{1}{p} + \frac{1}{q} + \frac{1}{r} \) where \( p, q, r \) are the altitudes of a triangle with area \( S \) and perimeter \( 2t \), we can follow these steps:
### Step 1: Understand the relationship between area, base, and height
The area \( S \) of a triangle can be expressed in terms of its base and height. For a triangle with base \( a \) and height \( p \) (altitude from vertex A), we have:
\[
S = \frac{1}{2} \times a \times p
\]
Similarly, we can express the area using the other sides and their respective altitudes:
\[
S = \frac{1}{2} \times b \times q
\]
\[
S = \frac{1}{2} \times c \times r
\]
### Step 2: Express the altitudes in terms of the area
From the area formulas, we can solve for the altitudes \( p, q, r \):
\[
p = \frac{2S}{a}, \quad q = \frac{2S}{b}, \quad r = \frac{2S}{c}
\]
### Step 3: Find the reciprocals of the altitudes
Now, we can find the reciprocals of the altitudes:
\[
\frac{1}{p} = \frac{a}{2S}, \quad \frac{1}{q} = \frac{b}{2S}, \quad \frac{1}{r} = \frac{c}{2S}
\]
### Step 4: Add the reciprocals
Now, we can add these reciprocals:
\[
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{a}{2S} + \frac{b}{2S} + \frac{c}{2S}
\]
This simplifies to:
\[
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{a + b + c}{2S}
\]
### Step 5: Substitute the perimeter
Since the perimeter of the triangle is given as \( 2t \), we have:
\[
a + b + c = 2t
\]
Thus, we can substitute this into our equation:
\[
\frac{1}{p} + \frac{1}{q} + \frac{1}{r} = \frac{2t}{2S} = \frac{t}{S}
\]
### Final Result
Therefore, the value of \( \frac{1}{p} + \frac{1}{q} + \frac{1}{r} \) is:
\[
\frac{t}{S}
\]
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