The circumcentre of the triangle lies within the triangle only when the triangle is :

Options

A. (a) Acute angled
B. (b) Right angled
C. (c) Isosceles
D. (d) None of these

Answer

To determine when the circumcenter of a triangle lies within the triangle, we can analyze the properties of different types of triangles: acute, right, and obtuse.

### Step-by-Step Solution:

1. **Understanding the Circumcenter**:
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. This point can be located inside or outside the triangle depending on the type of triangle.

**Hint**: Remember that the circumcenter is related to the perpendicular bisectors of the sides of the triangle.

2. **Acute Triangle**:
In an acute triangle, all angles are less than 90 degrees. When we draw the perpendicular bisectors of each side, they will intersect at a point that lies inside the triangle.

**Hint**: Visualize an acute triangle and draw its perpendicular bisectors to see where they meet.

3. **Right Triangle**:
In a right triangle, one angle is exactly 90 degrees. The circumcenter lies at the midpoint of the hypotenuse. Since the hypotenuse is one of the sides of the triangle, the circumcenter is on the boundary of the triangle, not inside it.

**Hint**: Consider the properties of a right triangle and how the circumcenter relates to the hypotenuse.

4. **Obtuse Triangle**:
In an obtuse triangle, one angle is greater than 90 degrees. The perpendicular bisectors of the sides will intersect at a point that lies outside the triangle.

**Hint**: Think about how the obtuse angle affects the position of the circumcenter relative to the triangle.

5. **Conclusion**:
The circumcenter of a triangle lies within the triangle only when the triangle is an **acute triangle**. Therefore, the correct answer to the question is that the circumcenter lies within the triangle only when the triangle is acute.

**Final Answer**: The circumcenter of the triangle lies within the triangle only when the triangle is **acute**.