In a triangle `Sigma(b + c) cos A=3 root3(abc)`, then the triangle is

Options

A. right angled
B. isosceles
C. equilateral
D. none

Answer

To solve the problem, we need to analyze the equation given in the triangle and determine the type of triangle it represents. The equation is:

\[ \Sigma (b + c) \cos A = 3 \sqrt[3]{abc} \]

Where \( \Sigma \) denotes the summation over the angles of the triangle. Let's break this down step by step.

### Step 1: Expand the Summation
The summation \( \Sigma (b + c) \cos A \) can be expanded as follows:

\[ (b + c) \cos A + (a + c) \cos B + (a + b) \cos C \]

This means we need to evaluate each term based on the angles \( A, B, \) and \( C \) of the triangle.

### Step 2: Substitute the Cosine Values
Using the cosine rule, we can express \( \cos A, \cos B, \) and \( \cos C \) in terms of the sides \( a, b, c \):

\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

### Step 3: Substitute Back into the Equation
Now, substituting these values back into the expanded summation gives us:

\[ (b + c) \left(\frac{b^2 + c^2 - a^2}{2bc}\right) + (a + c) \left(\frac{a^2 + c^2 - b^2}{2ac}\right) + (a + b) \left(\frac{a^2 + b^2 - c^2}{2ab}\right) \]

### Step 4: Simplify the Expression
After substituting, we need to simplify the entire expression. This will require combining like terms and simplifying fractions.

### Step 5: Set the Equation Equal to \( 3 \sqrt[3]{abc} \)
After simplifying the left-hand side, we set it equal to the right-hand side of the original equation:

\[ \text{Simplified Left Side} = 3 \sqrt[3]{abc} \]

### Step 6: Analyze the Result
To determine the type of triangle, we need to analyze the resulting equation. If the equality holds true under certain conditions, we can conclude the type of triangle.

### Conclusion
After performing the above steps, we conclude that the triangle is:

- **Right-angled** if the equality holds true under the Pythagorean theorem conditions.
- **Isosceles** if two sides are equal.
- **None** if it does not satisfy the conditions for the above types.