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

Answer

To solve the problem, we start with the given equation:

**Step 1: Analyze the given equation**
We have:
\[ a^2 + b^2 + c^2 = ab + bc + ca \]

**Step 2: Rearrange the equation**
Rearranging the equation gives us:
\[ a^2 + b^2 + c^2 - ab - bc - ca = 0 \]

**Step 3: Factor the expression**
This expression can be factored as:
\[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 0 \]

**Step 4: Understand the implications of the factorization**
Since the sum of squares is zero, each individual square must also be zero. Therefore:
\[ (a-b)^2 = 0 \]
\[ (b-c)^2 = 0 \]
\[ (c-a)^2 = 0 \]

This implies:
\[ a = b, \quad b = c, \quad c = a \]
Thus, we conclude that:
\[ a = b = c \]

**Step 5: Substitute into the expression we need to evaluate**
Now we need to evaluate the expression:
\[ \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} \]

Since \( a = b = c \), we can substitute \( a \) for \( b \) and \( c \):
\[ \frac{a+a}{a} + \frac{a+a}{a} + \frac{a+a}{a} \]

**Step 6: Simplify the expression**
This simplifies to:
\[ \frac{2a}{a} + \frac{2a}{a} + \frac{2a}{a} = 2 + 2 + 2 = 6 \]

**Final Answer:**
Thus, the value of the expression is:
\[ \boxed{6} \]

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