If `(x^(2))/(by+cz) = (y^(2))/(ax+ cz) = (z^(2))/(ax + by)=2`
`(x)/(x+2a) + (y)/(y+ 2b) + (z)/(z+ 2c)`= ?

Answer

To solve the problem, we start from the given equations:

1. \(\frac{x^2}{by + cz} = \frac{y^2}{ax + cz} = \frac{z^2}{ax + by} = 2\)

We can denote this common ratio as \(k\), where \(k = 2\).

From the first equation, we can express \(x^2\) in terms of \(y\) and \(z\):

\[ x^2 = 2(by + cz) \tag{1} \]

From the second equation, we express \(y^2\):

\[ y^2 = 2(ax + cz) \tag{2} \]

From the third equation, we express \(z^2\):

\[ z^2 = 2(ax + by) \tag{3} \]

Next, we need to find the value of:

\[ \frac{x}{x + 2a} + \frac{y}{y + 2b} + \frac{z}{z + 2c} \]

We will use the values of \(x^2\), \(y^2\), and \(z^2\) to express \(x\), \(y\), and \(z\) in terms of \(a\), \(b\), and \(c\).

### Step 1: Substitute the values of \(x^2\), \(y^2\), and \(z^2\)

Using equations (1), (2), and (3):

- From (1): \(x^2 = 2(by + cz)\)
- From (2): \(y^2 = 2(ax + cz)\)
- From (3): \(z^2 = 2(ax + by)\)

### Step 2: Find \(x\), \(y\), and \(z\)

Taking the square root of both sides, we have:

\[ x = \sqrt{2(by + cz)}, \quad y = \sqrt{2(ax + cz)}, \quad z = \sqrt{2(ax + by)} \]

### Step 3: Substitute \(x\), \(y\), and \(z\) into the expression

Now we substitute \(x\), \(y\), and \(z\) into the expression:

\[ \frac{\sqrt{2(by + cz)}}{\sqrt{2(by + cz)} + 2a} + \frac{\sqrt{2(ax + cz)}}{\sqrt{2(ax + cz)} + 2b} + \frac{\sqrt{2(ax + by)}}{\sqrt{2(ax + by)} + 2c} \]

### Step 4: Simplify each term

Each term can be simplified as follows:

\[ \frac{\sqrt{2(by + cz)}}{\sqrt{2(by + cz)} + 2a} = \frac{1}{1 + \frac{2a}{\sqrt{2(by + cz)}}} \]

Similarly for the other terms.

### Step 5: Combine the fractions

Now we can combine the fractions:

\[ \frac{1}{1 + \frac{2a}{\sqrt{2(by + cz)}}} + \frac{1}{1 + \frac{2b}{\sqrt{2(ax + cz)}}} + \frac{1}{1 + \frac{2c}{\sqrt{2(ax + by)}}} \]

### Final Result

After simplifying, we can find the common denominator and combine the fractions to get the final result.