In triangle ABC, the line joining the circumcenter and incenter is parallel to side AC, then `cos A + cos C` is equal to

Options

A. `(1)/(2)`
B. 1
C. `sqrt3`
D. 2

Explanation

To solve the problem step by step, we will analyze the given information and derive the required result.

### Step 1: Understand the Given Information
We are given a triangle ABC, where the line joining the circumcenter (O) and the incenter (I) is parallel to side AC. We need to find the value of \( \cos A + \cos C \).

### Step 2: Relate the Distances
Since the line joining the circumcenter and incenter is parallel to side AC, we can equate the distances from O and I to side AC.

- The distance from the circumcenter O to side AC is given by \( R \cos B \), where R is the circumradius.
- The distance from the incenter I to side AC is simply \( r \), where r is the inradius.

Since these distances are equal due to the parallel condition, we have:
\[ R \cos B = r \]

### Step 3: Use the Relationship Between R and r
From the relationship between the circumradius R and the inradius r in a triangle, we know:
\[ \frac{R}{r} = \frac{4 \Delta}{a^2} \]
where \( \Delta \) is the area of triangle ABC and \( a \) is the length of side BC.

### Step 4: Rearranging the Equation
From the equality \( R \cos B = r \), we can express \( \frac{R}{r} \) as:
\[ \frac{R}{r} = \frac{1}{\cos B} \]

### Step 5: Substitute into the Known Relationship
Using the known relationship \( \frac{R}{r} = \frac{4 \sin A}{\sin B \sin C} \), we can write:
\[ \frac{4 \sin A}{\sin B \sin C} = \frac{1}{\cos B} \]

### Step 6: Cross-Multiply and Simplify
Cross-multiplying gives us:
\[ 4 \sin A \cos B = \sin B \sin C \]

### Step 7: Use the Cosine Rule
Using the cosine rule in triangle ABC, we have:
\[ \cos A + \cos C = 1 - \cos B \]

### Step 8: Final Result
Substituting back into our previous equation, we find:
\[ \cos A + \cos C = 1 \]

Thus, the value of \( \cos A + \cos C \) is:
\[ \boxed{1} \]