A triangle PQR is formed by joining the mid-points of the sides of a triangle ABC. 'o' is the circumcentre of `triangle ABC `, then for `triangle PQR`, the point 'o' is :

Options

A. incentre
B. circumcentre
C. orthocentre
D. centroid

Answer

To solve the problem, we need to analyze the relationship between triangle ABC and triangle PQR, where P, Q, and R are the midpoints of the sides of triangle ABC. We also need to understand the properties of the circumcenter and orthocenter in relation to these triangles.

### Step-by-Step Solution:

1. **Identify Triangle ABC**:
Let's denote the vertices of triangle ABC as A, B, and C.

2. **Locate Midpoints**:
Find the midpoints of the sides of triangle ABC. Let:
- P be the midpoint of side AB,
- Q be the midpoint of side BC,
- R be the midpoint of side AC.

3. **Form Triangle PQR**:
Connect the midpoints P, Q, and R to form triangle PQR.

4. **Understand the Circumcenter**:
The circumcenter (O) of triangle ABC is the point where the perpendicular bisectors of the sides of triangle ABC intersect. This point is equidistant from all vertices A, B, and C.

5. **Relate Circumcenter to Triangle PQR**:
The circumcenter O of triangle ABC also has a relationship with triangle PQR. The circumcenter of triangle ABC is the same point that serves as the orthocenter of triangle PQR.

6. **Conclusion**:
Therefore, for triangle PQR, the point O (the circumcenter of triangle ABC) is the orthocenter of triangle PQR.

### Final Answer:
The point O is the orthocenter of triangle PQR.

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