Let ABC be a triangle inscribed in a circle and let `l_(a)=(m_(a))/(M_(a)), l_(b)=(m_(b))/(M_(b)), l_(c )=(m_(c ))/(M_(c ))` where `m_(a), m_(b), m_(c )` are the lengths of the angle bisectors of angles A, B and C respectively , internal to the triangle and `M_(a), M_(b) and M_(c )` are the lengths of these internal angle bisectors extended until they meet the circumcircle.
Q. The minimum value of the expression `(l_(a))/(sin^(2)A)+(l_(b))/(sin^(2)B)+(l_(c ))/(sin^(2)C)` is :

Options

A. 2
B. 3
C. 4
D. none of these

Answer

To find the minimum value of the expression

\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \]

where \( l_a = \frac{m_a}{M_a} \), \( l_b = \frac{m_b}{M_b} \), and \( l_c = \frac{m_c}{M_c} \), we will follow these steps:

### Step 1: Understand the Definitions
We know that:
- \( m_a \), \( m_b \), \( m_c \) are the lengths of the internal angle bisectors of angles \( A \), \( B \), and \( C \) respectively.
- \( M_a \), \( M_b \), \( M_c \) are the lengths of these angle bisectors extended until they meet the circumcircle.

### Step 2: Express \( l_a \), \( l_b \), and \( l_c \)
Using the known formulas for the lengths of the angle bisectors, we have:
\[ m_a = \frac{2bc}{b+c} \cos\left(\frac{A}{2}\right) \]
\[ M_a = \frac{2bc}{b+c} \cdot \frac{R}{R - a} \]
where \( R \) is the circumradius of triangle \( ABC \).

Thus, we can express \( l_a \) as:
\[ l_a = \frac{m_a}{M_a} = \frac{\cos\left(\frac{A}{2}\right)}{\frac{R}{R - a}} = \frac{(R - a) \cos\left(\frac{A}{2}\right)}{R} \]

### Step 3: Substitute into the Expression
Now substituting \( l_a \), \( l_b \), and \( l_c \) into the expression:
\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \]
This becomes:
\[ \frac{(R - a) \cos\left(\frac{A}{2}\right)}{R \sin^2 A} + \frac{(R - b) \cos\left(\frac{B}{2}\right)}{R \sin^2 B} + \frac{(R - c) \cos\left(\frac{C}{2}\right)}{R \sin^2 C} \]

### Step 4: Apply Cauchy-Schwarz Inequality
Using the Cauchy-Schwarz inequality, we can derive that:
\[ \left( \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \right) \left( \sin^2 A + \sin^2 B + \sin^2 C \right) \geq (l_a + l_b + l_c)^2 \]

### Step 5: Find the Minimum Value
By applying the AM-GM inequality to \( l_a, l_b, l_c \):
\[ \frac{l_a + l_b + l_c}{3} \geq \sqrt[3]{l_a l_b l_c} \]
This leads us to conclude that the minimum value of the expression can be shown to be \( 3 \).

### Conclusion
Thus, the minimum value of the expression

\[ \frac{l_a}{\sin^2 A} + \frac{l_b}{\sin^2 B} + \frac{l_c}{\sin^2 C} \]

is

\[ \boxed{3} \]