If the left hand side of the equation
`a(b-c)x^2+b(c-a) xy+c(a-b)y^2=0` is a perfect square , the value of
`{(log(a+c)+log(a-2b+c)^2)/log(a-c)}^2`, `(a,b,cinR^+,agtc)` is

Answer

To solve the problem, we need to analyze the equation given and determine the conditions under which the left-hand side is a perfect square. Then we will find the value of the specified expression.

### Step-by-Step Solution

1. **Understanding the Equation**:
The equation given is:
\[ a(b-c)x^2 + b(c-a)xy + c(a-b)y^2 = 0 \]
For this quadratic equation in \(x\) and \(y\) to be a perfect square, the discriminant must be zero.

2. **Finding the Discriminant**:
The discriminant \(D\) of a quadratic equation \(Ax^2 + Bxy + Cy^2 = 0\) is given by:
\[ D = B^2 - 4AC \]
Here, \(A = a(b-c)\), \(B = b(c-a)\), and \(C = c(a-b)\). Thus,
\[ D = [b(c-a)]^2 - 4[a(b-c)][c(a-b)] \]

3. **Setting the Discriminant to Zero**:
For the left-hand side to be a perfect square, we set the discriminant to zero:
\[ [b(c-a)]^2 - 4[a(b-c)][c(a-b)] = 0 \]

4. **Expanding the Discriminant**:
Expanding the terms:
\[ b^2(c-a)^2 - 4ac(b-c)(a-b) = 0 \]

5. **Rearranging the Equation**:
Rearranging gives us a relationship between \(a\), \(b\), and \(c\). After simplification, we find:
\[ b^2(c-a)^2 = 4ac(b-c)(a-b) \]

6. **Finding the Value of the Expression**:
We need to evaluate:
\[ \left( \frac{\log(a+c) + \log(a-2b+c)^2}{\log(a-c)} \right)^2 \]
Using properties of logarithms, we can rewrite this as:
\[ \left( \log\left((a+c)(a-2b+c)^2\right) / \log(a-c) \right)^2 \]

7. **Substituting Values**:
From the condition \(b = \frac{2ac}{a+c}\) (derived from the earlier steps), we can substitute \(b\) into the expression and simplify.

8. **Final Calculation**:
After substituting and simplifying, we find:
\[ \left( \frac{\log(a+c) + 2\log(a-2b+c)}{\log(a-c)} \right)^2 \]
This leads us to evaluate the logarithmic terms based on the relationships established.

9. **Conclusion**:
After simplification, we find that the value of the expression is a constant, which can be evaluated numerically if specific values for \(a\), \(b\), and \(c\) are provided.