O is orthocentre of a triangle PQR, which is formed by joining the mid- points of the sides of a `triangle ABC`, is :

Options

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

Answer

To solve the problem, we need to understand the relationship between the orthocenter of triangle PQR and triangle ABC. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the triangles
We have triangle ABC, and we are given that triangle PQR is formed by joining the midpoints of the sides of triangle ABC.

### Step 2: Determine the midpoints
Let’s denote the midpoints of sides AB, BC, and CA as follows:
- Let M be the midpoint of side AB.
- Let N be the midpoint of side BC.
- Let O be the midpoint of side CA.

Thus, triangle PQR is formed by the points M, N, and O.

### Step 3: Understand the orthocenter
The orthocenter of a triangle is the point where the three altitudes intersect. For triangle PQR, we denote this orthocenter as point O.

### Step 4: Relate the orthocenter of triangle PQR to triangle ABC
When we consider triangle ABC, the altitudes from vertices A, B, and C will intersect at a point, which is also the orthocenter of triangle ABC.

### Step 5: Establish the relationship
It is a known geometric property that the orthocenter of triangle ABC is related to the orthocenter of triangle PQR. Specifically, the orthocenter of triangle PQR (point O) is also the orthocenter of triangle ABC.

### Conclusion
Thus, the orthocenter of triangle PQR is the same point as the orthocenter of triangle ABC.

### Final Answer
The orthocenter O of triangle PQR is also the orthocenter of triangle ABC.

---