D,E and F are the middle points of the sides of the triangle ABC, then

Options

A. centroid of the triangle DEF is the same as that of ABC
B. orthocentre of the triangle DEF is the circumcentre of ABC
C. orthocentre of the triangle DEF is the incentre of ABC
D. centroid of the triangle DEF is not the same as that of ABC

Answer

To solve the problem, we need to analyze the triangle ABC and its midpoints D, E, and F, and then verify the statements regarding the centroid, orthocenter, and incenter of the triangles formed.

### Step-by-Step Solution:

1. **Identify the Midpoints**:
- Let D, E, and F be the midpoints of sides BC, AC, and AB of triangle ABC, respectively.

2. **Draw Triangle ABC**:
- Draw triangle ABC with vertices A, B, and C.

3. **Construct Triangle DEF**:
- Connect points D, E, and F to form triangle DEF.

4. **Centroid of Triangle ABC**:
- The centroid (G) of triangle ABC is the point where the three medians intersect. A median connects a vertex to the midpoint of the opposite side.

5. **Centroid of Triangle DEF**:
- Similarly, the centroid (G') of triangle DEF is the point where the medians of triangle DEF intersect.

6. **Relationship Between Centroids**:
- Since D, E, and F are midpoints of the sides of triangle ABC, the centroid G of triangle ABC and the centroid G' of triangle DEF coincide. Therefore, the centroid of triangle DEF is the same as that of triangle ABC.

7. **Orthocenter of Triangle DEF**:
- The orthocenter (O) of triangle DEF is the point where the altitudes of triangle DEF intersect.

8. **Circumcenter of Triangle ABC**:
- The circumcenter (O') of triangle ABC is the point where the perpendicular bisectors of the sides of triangle ABC intersect.

9. **Relationship Between Orthocenter and Circumcenter**:
- It can be shown that the orthocenter O of triangle DEF is the same point as the circumcenter O' of triangle ABC.

10. **Incenter of Triangle ABC**:
- The incenter (I) of triangle ABC is the point where the angle bisectors of triangle ABC intersect.

11. **Relationship Between Orthocenter and Incenter**:
- The orthocenter O of triangle DEF coincides with the incenter I of triangle ABC.

12. **Final Verification**:
- We conclude that:
- The centroid of triangle DEF is the same as that of triangle ABC.
- The orthocenter of triangle DEF is the circumcenter of triangle ABC.
- The orthocenter of triangle DEF is also the incenter of triangle ABC.
- Therefore, the statement that the centroid of triangle DEF is not the same as that of triangle ABC is incorrect.

### Summary of Results:
- Centroid of triangle DEF = Centroid of triangle ABC (True)
- Orthocenter of triangle DEF = Circumcenter of triangle ABC (True)
- Orthocenter of triangle DEF = Incenter of triangle ABC (True)
- Centroid of triangle DEF ≠ Centroid of triangle ABC (False)