If l, m, n denote the side of a pedal triangle, then `(l)/(a ^(2))+(m)/(b^(2))+(n)/(c ^(2))` is equal to

Options

A. (a)`(a^(2) +b^(2)+c^(2))/(a ^(3)+b ^(3) +c^(2))`
B. (b)`(a^(2)+b ^(2)+c^(2))/(2abc)`
C. (c)`(a^(3)+ b ^(3)+ c^(3))/(abc(a+b+c))`
D. (d)`1/a+1/b+1/c`

Answer

To solve the problem, we need to find the value of the expression \(\frac{l}{a^2} + \frac{m}{b^2} + \frac{n}{c^2}\) where \(l\), \(m\), and \(n\) are the sides of a pedal triangle corresponding to the angles \(A\), \(B\), and \(C\) of the original triangle with sides \(a\), \(b\), and \(c\).

### Step-by-Step Solution:

1. **Understanding the Pedal Triangle**:
The pedal triangle is formed by dropping perpendiculars from a point (often the orthocenter or circumcenter) to the sides of the triangle. The lengths of these perpendiculars correspond to the sides of the pedal triangle.

2. **Expressing the Sides of the Pedal Triangle**:
The sides of the pedal triangle can be expressed in terms of the original triangle's sides and angles:
\[ l = a \cos A, \quad m = b \cos B, \quad n = c \cos C \]

3. **Substituting into the Expression**:
We substitute \(l\), \(m\), and \(n\) into the expression:
\[ \frac{l}{a^2} + \frac{m}{b^2} + \frac{n}{c^2} = \frac{a \cos A}{a^2} + \frac{b \cos B}{b^2} + \frac{c \cos C}{c^2} \]

4. **Simplifying Each Term**:
Each term simplifies as follows:
\[ \frac{l}{a^2} = \frac{\cos A}{a}, \quad \frac{m}{b^2} = \frac{\cos B}{b}, \quad \frac{n}{c^2} = \frac{\cos C}{c} \]
Therefore, the expression becomes:
\[ \frac{\cos A}{a} + \frac{\cos B}{b} + \frac{\cos C}{c} \]

5. **Finding a Common Denominator**:
To combine the terms, we can find a common denominator, which is \(abc\):
\[ \frac{b c \cos A + a c \cos B + a b \cos C}{abc} \]

6. **Using the Cosine Rule**:
We can use the cosine rule to express \(b c \cos A + a c \cos B + a b \cos C\):
\[ b c \cos A = \frac{a^2 + b^2 - c^2}{2} \]
\[ a c \cos B = \frac{a^2 + c^2 - b^2}{2} \]
\[ a b \cos C = \frac{b^2 + c^2 - a^2}{2} \]

7. **Combining the Results**:
Adding these three equations together gives:
\[ b c \cos A + a c \cos B + a b \cos C = \frac{(a^2 + b^2 - c^2) + (a^2 + c^2 - b^2) + (b^2 + c^2 - a^2)}{2} \]
This simplifies to:
\[ = \frac{2(a^2 + b^2 + c^2)}{2} = a^2 + b^2 + c^2 \]

8. **Final Expression**:
Thus, we have:
\[ \frac{l}{a^2} + \frac{m}{b^2} + \frac{n}{c^2} = \frac{a^2 + b^2 + c^2}{abc} \]

### Conclusion:
The final result is:
\[ \frac{l}{a^2} + \frac{m}{b^2} + \frac{n}{c^2} = \frac{a^2 + b^2 + c^2}{abc} \]