Find the distance between the circumcentre and the incentre of the `DeltaABC.`

Answer

To find the distance between the circumcenter (denoted as \( O \)) and the incenter (denoted as \( I \)) of triangle \( ABC \), we can use the following steps:

### Step-by-Step Solution:

1. **Identify the Points**:
- Let \( O \) be the circumcenter of triangle \( ABC \).
- Let \( I \) be the incenter of triangle \( ABC \).

2. **Draw the Relevant Lines**:
- Extend the line segment \( OI \) to meet the circumcircle of triangle \( ABC \) at points \( X \) and \( Y \).

3. **Draw the Angle Bisectors**:
- Draw the angle bisectors \( AI \) and \( BI \) of angles \( A \) and \( B \) respectively.

4. **Extend the Angle Bisector**:
- Extend the angle bisector \( AI \) to meet the circumcircle at point \( P \).

5. **Join Points**:
- Join points \( P \) and \( O \) to form line segment \( PO \). Extend this line to meet the circumcircle at point \( Q \).

6. **Use the Intersecting Chords Theorem**:
- According to the theorem, if two chords \( AP \) and \( XY \) intersect at point \( I \), then:
\[ AI \cdot IP = XI \cdot IY \]

7. **Define Radii**:
- Let \( R \) be the circumradius (distance from \( O \) to \( X \) and \( Y \)).
- Define \( XI = R - d \) and \( IY = R + d \), where \( d \) is the distance \( OI \).

8. **Substitute into the Intersecting Chords Equation**:
- Substitute the expressions for \( XI \) and \( IY \) into the intersecting chords equation:
\[ AI \cdot IP = (R - d)(R + d) = R^2 - d^2 \]

9. **Use Properties of Angles**:
- In triangle \( AIB \), the angles \( \angle AIB = \frac{A}{2} + \frac{B}{2} = \frac{C}{2} \).

10. **Establish Similar Triangles**:
- By drawing a perpendicular from \( I \) to \( AB \), we can establish that triangles \( AIL \) and \( QBP \) are similar by AA similarity criterion.

11. **Set Up Ratios from Similar Triangles**:
- From the similarity, we have:
\[ \frac{AI}{QP} = \frac{LI}{BP} \]

12. **Substitute Values**:
- Substitute \( LI = r \) (the inradius) and \( QP = 2R \):
\[ AI \cdot BP = r \cdot 2R \]

13. **Combine Equations**:
- Substitute \( AI \cdot BP \) into the equation from step 8:
\[ r \cdot 2R = R^2 - d^2 \]

14. **Solve for \( d^2 \)**:
- Rearranging gives:
\[ d^2 = R^2 - 2rR \]

15. **Final Distance Formula**:
- Thus, the distance \( d \) between the circumcenter and incenter is:
\[ d = \sqrt{R^2 - 2rR} \]