Let x,y an z be positive real numbers. Suppose x, y and z are the length of the sides of triangle opposite ot its angles, X,Y and Z respectively.If
`"tan" (X)/(2)+"tan"(Z)/(2)=(2y)/(x+y+x)` then which of the following statement is/are TURE ?

Options

A. `2Y=X+Z`
B. `Y=X+Z`
C. `"tan"(X)/(2)=(x)/(y+z)`
D. `x^(2)+z^(2)-y^(2)=xz`

Answer

To solve the problem, we need to analyze the given equation involving the sides of a triangle and the angles opposite to those sides. The equation provided is:

\[ \frac{\tan(X/2) + \tan(Z/2)}{2} = \frac{2y}{x + y + z} \]

### Step 1: Use the formula for \(\tan(A/2)\)

We know that for any angle \(A\) in a triangle, the half-angle tangent can be expressed as:

\[ \tan\left(\frac{A}{2}\right) = \frac{\Delta}{s(s-a)} \]

where \(\Delta\) is the area of the triangle, \(s\) is the semi-perimeter, and \(a\) is the side opposite angle \(A\).

### Step 2: Apply the formula to our angles

For our triangle with sides \(x\), \(y\), and \(z\) opposite angles \(X\), \(Y\), and \(Z\) respectively, we can write:

\[ \tan\left(\frac{X}{2}\right) = \frac{\Delta}{s(s-x)} \quad \text{and} \quad \tan\left(\frac{Z}{2}\right) = \frac{\Delta}{s(s-z)} \]

### Step 3: Substitute into the equation

Substituting these into our original equation gives:

\[ \frac{\frac{\Delta}{s(s-x)} + \frac{\Delta}{s(s-z)}}{2} = \frac{2y}{x + y + z} \]

### Step 4: Simplify the left-hand side

Combining the terms on the left-hand side:

\[ \frac{\Delta \left( \frac{1}{s(s-x)} + \frac{1}{s(s-z)} \right)}{2} = \frac{\Delta \left( \frac{s-z + s-x}{s(s-x)(s-z)} \right)}{2} \]

This simplifies to:

\[ \frac{\Delta (2s - x - z)}{2s(s-x)(s-z)} \]

### Step 5: Set the equation equal to the right-hand side

Now we have:

\[ \frac{\Delta (2s - x - z)}{2s(s-x)(s-z)} = \frac{2y}{x + y + z} \]

### Step 6: Cross-multiply to eliminate fractions

Cross-multiplying gives:

\[ \Delta (2s - x - z)(x + y + z) = 4sy(s-x)(s-z) \]

### Step 7: Analyze the implications

This equation relates the area of the triangle to its sides and can be used to derive relationships between the angles and sides.

### Step 8: Determine the conditions for the triangle

From the analysis, we can derive that:

1. If \(y\) is the longest side, then \(y^2 = x^2 + z^2\) implies that angle \(Y\) is \(90^\circ\).
2. Thus, \(X + Z = 90^\circ\).

### Conclusion

After analyzing the conditions, we can conclude that:

- Option B: \(y = x + z\) is true.
- Option C: \(\tan(X/2) = \frac{x}{y + z}\) is also true.

### Final Answer

The correct statements are B and C.

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