In `Delta ABC`, the ratio `a/sin A=b/sinB=c/sinC` is not always equal to (All symbols used have usual meaning in a triangle.)

Options

A. 2R, where R is the circumradius
B. `(abc)/(2Delta),` where `Delta` is the area of the triangle
C. `2/3 (a^(2)+ b^(2) +c^(2)) ^(1/2)`
D. `((abc)^(2/3))/((h_(1)h_(2) h_(3))^(1/3)`

Answer

To solve the problem, we need to show that the ratio \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) is not always equal to \( 2R \) or any other specific expression, except for certain cases. Here’s a step-by-step solution:

### Step 1: Understand the Law of Sines
The Law of Sines states that for any triangle \( ABC \):
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

### Step 2: Identify the Circumradius
The circumradius \( R \) can be expressed in terms of the area \( \Delta \) of the triangle:
\[ R = \frac{abc}{4\Delta} \]
Thus, we can rewrite the Law of Sines as:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = \frac{abc}{2\Delta} \]

### Step 3: Analyze the Expression
We need to analyze if \( \frac{a}{\sin A} \) can be equal to other expressions such as \( \frac{abc}{2\Delta} \) or \( 2R \) in all cases.

### Step 4: Check Specific Cases
1. **Equilateral Triangle**: In an equilateral triangle, all angles are \( 60^\circ \) and all sides are equal. Here, the ratio holds true.
2. **Isosceles Triangle**: In an isosceles triangle, the ratio holds true for the equal sides but may not hold for the unequal side.
3. **Scalene Triangle**: In a scalene triangle, the ratio may not hold true as the angles and sides vary.

### Step 5: Conclusion
The ratio \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) is not always equal to \( 2R \) or any other specific expression like \( \frac{abc}{2\Delta} \) in all triangles. It is only valid under certain conditions (like in an equilateral triangle).

### Final Answer
Thus, the statement that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \) is not always equal to \( 2R \) or any other specific expression is true.

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