If `p_(1),p_(2),p_(3)` are altitudes of a triangle ABC from the vertices A,B,C and `triangle` the area of the triangle, then `p_(1)^(-2)+p_(2)^(-2)+p_(3)^(-2)` is equal to

Options

A. `(a+b+c)/(Delta)`
B. `(a^(2)+b^(2)+c^(2))/(4Delta^(2))`
C. `(a^(2)+b^(2)+c^(2))/(Delta^(2))`
D. none of these

Answer

To solve the problem, we need to find the value of \( p_1^{-2} + p_2^{-2} + p_3^{-2} \), where \( p_1, p_2, p_3 \) are the altitudes of triangle \( ABC \) from vertices \( A, B, C \) respectively, and \( \Delta \) is the area of the triangle.

### Step-by-Step Solution:

1. **Understanding the Area of the Triangle**:
The area \( \Delta \) of triangle \( ABC \) can be expressed in terms of its sides and corresponding altitudes:
\[ \Delta = \frac{1}{2} \times \text{base} \times \text{height} \]

2. **Finding Altitudes**:
- For altitude \( p_1 \) from vertex \( A \) to side \( BC \) (length \( a \)):
\[ \Delta = \frac{1}{2} \times a \times p_1 \implies p_1 = \frac{2\Delta}{a} \]
- For altitude \( p_2 \) from vertex \( B \) to side \( AC \) (length \( b \)):
\[ \Delta = \frac{1}{2} \times b \times p_2 \implies p_2 = \frac{2\Delta}{b} \]
- For altitude \( p_3 \) from vertex \( C \) to side \( AB \) (length \( c \)):
\[ \Delta = \frac{1}{2} \times c \times p_3 \implies p_3 = \frac{2\Delta}{c} \]

3. **Calculating \( p_1^{-2}, p_2^{-2}, p_3^{-2} \)**:
- Now we compute the squares of the altitudes:
\[ p_1^{-2} = \left(\frac{2\Delta}{a}\right)^{-2} = \frac{a^2}{4\Delta^2} \]
\[ p_2^{-2} = \left(\frac{2\Delta}{b}\right)^{-2} = \frac{b^2}{4\Delta^2} \]
\[ p_3^{-2} = \left(\frac{2\Delta}{c}\right)^{-2} = \frac{c^2}{4\Delta^2} \]

4. **Summing the Inverse Squares**:
- Now we add these values together:
\[ p_1^{-2} + p_2^{-2} + p_3^{-2} = \frac{a^2}{4\Delta^2} + \frac{b^2}{4\Delta^2} + \frac{c^2}{4\Delta^2} \]
- This can be simplified:
\[ p_1^{-2} + p_2^{-2} + p_3^{-2} = \frac{a^2 + b^2 + c^2}{4\Delta^2} \]

5. **Final Result**:
- Therefore, the final expression is:
\[ p_1^{-2} + p_2^{-2} + p_3^{-2} = \frac{a^2 + b^2 + c^2}{4\Delta^2} \]

### Conclusion:
The value of \( p_1^{-2} + p_2^{-2} + p_3^{-2} \) is equal to \( \frac{a^2 + b^2 + c^2}{4\Delta^2} \).