Given vectors `barCB=bara, barCA=barb and barCO=barx` where O is the centre of circle circumscribed about `DeltaABC`, then find vector `barx`

Answer

To solve the problem of finding the vector \(\bar{x}\) given the vectors \(\bar{CB} = \bar{a}\), \(\bar{CA} = \bar{b}\), and \(\bar{CO} = \bar{x}\), where \(O\) is the center of the circle circumscribed about triangle \(ABC\), we can follow these steps:

### Step 1: Write down the relationships between the vectors
From the triangle law of vector addition, we have:
\[ \bar{CA} + \bar{AB} = \bar{CB} \]
Substituting the given vectors:
\[ \bar{b} + \bar{AB} = \bar{a} \]
From this, we can express \(\bar{AB}\) as:
\[ \bar{AB} = \bar{a} - \bar{b} \tag{1} \]

### Step 2: Use the relationship involving the circumcenter
We know that the circumcenter \(O\) satisfies:
\[ \bar{CO} + \bar{OA} = \bar{CA} \]
Substituting the vectors:
\[ \bar{x} + \bar{OA} = \bar{b} \]
From this, we can express \(\bar{OA}\) as:
\[ \bar{OA} = \bar{b} - \bar{x} \tag{2} \]

### Step 3: Use another relationship involving the circumcenter
Similarly, for the vector \(CB\):
\[ \bar{CO} + \bar{OB} = \bar{CB} \]
Substituting the vectors:
\[ \bar{x} + \bar{OB} = \bar{a} \]
From this, we can express \(\bar{OB}\) as:
\[ \bar{OB} = \bar{a} - \bar{x} \tag{3} \]

### Step 4: Equate the distances from the circumcenter to the vertices
Since \(O\) is the circumcenter, the distances from \(O\) to \(A\), \(B\), and \(C\) are equal:
\[ |\bar{OA}| = |\bar{OB}| = |\bar{OC}| = r \]
This implies:
\[ |\bar{b} - \bar{x}| = |\bar{a} - \bar{x}| = |\bar{x}| \]

### Step 5: Set up the equations based on the magnitudes
From the above equalities, we can write:
1. \(|\bar{b} - \bar{x}| = r\)
2. \(|\bar{a} - \bar{x}| = r\)
3. \(|\bar{x}| = r\)

### Step 6: Solve for \(\bar{x}\)
From the equations, we can derive:
- From (1) and (3):
\[ \bar{b} - \bar{x} = r \quad \text{and} \quad \bar{x} = r \]
This leads to:
\[ \bar{b} - r = r \implies \bar{b} = 2r \]

- From (2) and (3):
\[ \bar{a} - \bar{x} = r \quad \text{and} \quad \bar{x} = r \]
This leads to:
\[ \bar{a} - r = r \implies \bar{a} = 2r \]

### Final Result
Thus, we find:
\[ \bar{x} = \frac{\bar{a} + \bar{b}}{2} \]