In a `Delta XYZ ` , O, T, C are orthocentre , incentre and circumcentre of this triangle `angle`XOZ = 2.5 `angle`XCZ Find `angle`XIZ

Answer

To find the angle \( \angle XIZ \) in triangle \( XYZ \) where \( O \) is the orthocenter, \( I \) is the incenter, and \( C \) is the circumcenter, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- We are given that \( \angle XOZ = 2.5 \times \angle XCZ \).
- We need to find \( \angle XIZ \).

2. **Relate Angles**:
- From the properties of the circumcenter, we know that:
\[ \angle XCZ = 2 \times \angle Y \]
- From the properties of the orthocenter, we have:
\[ \angle XOZ = 180^\circ - \angle Y \]

3. **Set Up the Equation**:
- Substitute the expressions for \( \angle XOZ \) and \( \angle XCZ \) into the given equation:
\[ 180^\circ - \angle Y = 2.5 \times (2 \times \angle Y) \]
- This simplifies to:
\[ 180^\circ - \angle Y = 5 \times \angle Y \]

4. **Combine Like Terms**:
- Rearranging the equation gives:
\[ 180^\circ = 5 \times \angle Y + \angle Y \]
- Which simplifies to:
\[ 180^\circ = 6 \times \angle Y \]

5. **Solve for \( \angle Y \)**:
- Dividing both sides by 6:
\[ \angle Y = \frac{180^\circ}{6} = 30^\circ \]

6. **Find \( \angle XIZ \)**:
- The angle \( \angle XIZ \) can be found using the formula:
\[ \angle XIZ = 90^\circ + \frac{\angle Y}{2} \]
- Substituting \( \angle Y = 30^\circ \):
\[ \angle XIZ = 90^\circ + \frac{30^\circ}{2} = 90^\circ + 15^\circ = 105^\circ \]

### Final Answer:
Thus, the value of \( \angle XIZ \) is \( 105^\circ \).

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