If in a `DeltaABC,'S'` is the circumcentre then :

Options

A. S is equidistant from all the vertices of a triangle
B. S is equidistant from all the sides of a triangle
C. AS, BS and CS are the angular bisectors
D. AS, BS and CS produced are the altitudes on the opposite sides.

Answer

To solve the question regarding the circumcenter \( S \) of triangle \( ABC \), we will analyze the properties of the circumcenter and its relationship with the vertices of the triangle.

### Step-by-Step Solution:

1. **Understanding the Circumcenter**:
- The circumcenter \( S \) of a triangle is defined as the point where the perpendicular bisectors of the sides of the triangle intersect. It is also the center of the circumcircle, which is the circle that passes through all three vertices of the triangle.

**Hint**: Remember that the circumcenter is equidistant from all vertices of the triangle.

2. **Equidistance from Vertices**:
- Since \( S \) is the circumcenter, it is equidistant from all three vertices \( A \), \( B \), and \( C \). This means that the distances \( SA \), \( SB \), and \( SC \) are equal. We can denote this common distance as \( R \) (the circumradius).

**Hint**: Visualize the circumcircle; all points on the circle are at the same distance from the center.

3. **Circumcircle**:
- The circumcircle is the circle that passes through points \( A \), \( B \), and \( C \). The radius of this circle is the distance from the circumcenter \( S \) to any vertex of the triangle.

**Hint**: The circumcircle's radius is crucial in understanding the relationship between the circumcenter and the triangle's vertices.

4. **Angle Bisectors**:
- The segments \( AS \), \( BS \), and \( CS \) are not necessarily the angle bisectors of triangle \( ABC \). The angle bisectors are segments that divide the angles at vertices \( A \), \( B \), and \( C \) into two equal parts, and they intersect at the incenter, not the circumcenter.

**Hint**: Differentiate between angle bisectors and segments connecting the circumcenter to the vertices.

5. **Altitude**:
- The segments \( AS \), \( BS \), and \( CS \) are not produced to the altitudes on the opposite sides. The altitudes of a triangle are the perpendicular segments from each vertex to the line containing the opposite side, and they intersect at the orthocenter, which is different from the circumcenter.

**Hint**: Recall the definitions of altitudes and how they relate to the triangle's vertices and sides.

### Conclusion:
The correct statement regarding the circumcenter \( S \) is that it is equidistant from all the vertices of triangle \( ABC \). The other statements regarding angular bisectors and altitudes are incorrect.