If in a `Delta ABC, cos A+2 cosB+cos C=2, " then " a,b,c` are in

Options

A. A.P.
B. G.P.
C. H.P.
D. None

Answer

To solve the problem, we need to analyze the equation given in the triangle ABC:

**Given:**
\[ \cos A + 2 \cos B + \cos C = 2 \]

**Step 1: Rearranging the equation**
We can rearrange the equation to isolate the cosine terms:
\[ \cos A + \cos C = 2 - 2 \cos B \]

**Hint:** Rearranging the equation helps to simplify the expression and isolate terms.

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**Step 2: Applying the Cosine Rule**
Using the cosine rule, we can express \(\cos A\), \(\cos B\), and \(\cos C\) in terms of the sides \(a\), \(b\), and \(c\) of the triangle:
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

**Hint:** The cosine rule relates the angles of a triangle to the lengths of its sides.

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**Step 3: Substitute the cosine values**
Substituting these values into the rearranged equation:
\[ \frac{b^2 + c^2 - a^2}{2bc} + \frac{a^2 + b^2 - c^2}{2ab} = 2 - 2 \left(\frac{a^2 + c^2 - b^2}{2ac}\right) \]

**Hint:** Substituting the cosine values allows us to express everything in terms of the sides of the triangle.

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**Step 4: Finding a common denominator**
To simplify the left-hand side, we find a common denominator:
\[ \frac{(b^2 + c^2 - a^2) \cdot a + (a^2 + b^2 - c^2) \cdot c}{2abc} = 2 - \frac{(a^2 + c^2 - b^2)}{ac} \]

**Hint:** Finding a common denominator is crucial for combining fractions.

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**Step 5: Simplifying the equation**
Now, simplifying both sides leads to:
\[ (b^2 + c^2 - a^2) \cdot a + (a^2 + b^2 - c^2) \cdot c = 2abc - (a^2 + c^2 - b^2) \cdot 2b \]

**Hint:** Simplifying the equation step-by-step helps in identifying relationships between the sides.

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**Step 6: Rearranging terms**
Rearranging the equation gives us:
\[ b^2 + c^2 + a^2 + 2bc = 2ab + 2ac \]

**Hint:** Rearranging terms can reveal patterns or relationships.

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**Step 7: Analyzing the result**
From the rearranged equation, we can see that:
\[ b^2 + c^2 = 2ab + 2ac - a^2 \]
This suggests that the sides \(a\), \(b\), and \(c\) have a specific relationship.

**Hint:** Analyzing the equation can lead to conclusions about the nature of the triangle.

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**Step 8: Conclusion**
From the derived relationships, we can conclude that:
\[ b - a = c - b \]
This implies that \(a\), \(b\), and \(c\) are in an arithmetic progression (AP).

**Final Answer:**
Thus, the sides \(a\), \(b\), and \(c\) are in an arithmetic progression (AP).

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