In a triangle ABC,
abc s `sin "" A/2 sin ""B/2 sin "" C/2=`

Options

A. `Delta^2`
B. `3Delta^2`
C. `Delta^3`
D. none

Answer

To solve the problem, we need to find the value of the expression:

\[ abc \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) \]

where \( A, B, C \) are the angles of triangle \( ABC \) and \( a, b, c \) are the lengths of the sides opposite to these angles respectively.

### Step 1: Use the half-angle sine formulas

The half-angle sine formulas for a triangle are given by:

\[ \sin\left(\frac{A}{2}\right) = \sqrt{\frac{s(s-a)}{bc}}, \quad \sin\left(\frac{B}{2}\right) = \sqrt{\frac{s(s-b)}{ac}}, \quad \sin\left(\frac{C}{2}\right) = \sqrt{\frac{s(s-c)}{ab}} \]

where \( s \) is the semi-perimeter of the triangle defined as:

\[ s = \frac{a + b + c}{2} \]

### Step 2: Substitute the half-angle sine formulas into the expression

Now we substitute these formulas into the original expression:

\[ abc \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) = abc \cdot \sqrt{\frac{s(s-a)}{bc}} \cdot \sqrt{\frac{s(s-b)}{ac}} \cdot \sqrt{\frac{s(s-c)}{ab}} \]

### Step 3: Simplify the expression

Combining the square roots:

\[ = abc \cdot \sqrt{\frac{s(s-a) \cdot s(s-b) \cdot s(s-c)}{(bc)(ac)(ab)}} \]

This simplifies to:

\[ = abc \cdot \sqrt{\frac{s^3(s-a)(s-b)(s-c)}{a^2b^2c^2}} \]

### Step 4: Factor out common terms

Now we can factor out \( abc \):

\[ = \sqrt{s^3(s-a)(s-b)(s-c)} \cdot \frac{abc}{abc} = \sqrt{s^3(s-a)(s-b)(s-c)} \]

### Step 5: Relate to the area of triangle

The area \( K \) of triangle \( ABC \) can be expressed using Heron's formula as:

\[ K = \sqrt{s(s-a)(s-b)(s-c)} \]

Thus, we have:

\[ abc \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) = \sqrt{s^3} \cdot K \]

### Final Result

The final expression simplifies to:

\[ abc \cdot \sin\left(\frac{A}{2}\right) \cdot \sin\left(\frac{B}{2}\right) \cdot \sin\left(\frac{C}{2}\right) = \frac{K \cdot s^{3/2}}{abc} \]