If `a^(2) + b^(2) + c^(2) =ab + bc + ca` find `(a + c)/(b)`= ?

Answer

To solve the equation \( a^2 + b^2 + c^2 = ab + bc + ca \) and find the value of \( \frac{a+c}{b} \), we can follow these steps:

### Step 1: Rearrange the equation
Start by rearranging the given equation:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]

### Step 2: Multiply the entire equation by 2
To simplify the expression, multiply the entire equation by 2:
\[ 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0 \]

### Step 3: Group terms
Rearranging the terms gives:
\[ 2a^2 - 2ab + 2b^2 - 2bc + 2c^2 - 2ca = 0 \]

### Step 4: Recognize the quadratic form
This can be recognized as:
\[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \]
This means that each squared term must equal zero since squares are always non-negative.

### Step 5: Set each squared term to zero
From the equation \( (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \), we conclude:
\[ a - b = 0 \quad \Rightarrow \quad a = b \]
\[ b - c = 0 \quad \Rightarrow \quad b = c \]
\[ c - a = 0 \quad \Rightarrow \quad c = a \]

### Step 6: Conclude that all variables are equal
Thus, we find that:
\[ a = b = c \]

### Step 7: Substitute back to find \( \frac{a+c}{b} \)
Now, substituting \( c = a \) into \( \frac{a+c}{b} \):
\[ \frac{a+c}{b} = \frac{a+a}{a} = \frac{2a}{a} = 2 \]

### Final Answer
Thus, the value of \( \frac{a+c}{b} \) is:
\[ \boxed{2} \]