If AD, BE, CF are the altitudes of `DeltaABC` whose orthocentre is H, then C is the orthocentre of :

Options

A. `DeltaABH`
B. `DeltaBDH`
C. `DeltaABD`
D. `DeltaBEA`

Answer

To determine which triangle has C as its orthocenter, we need to analyze the properties of the orthocenter in relation to triangle ABC and its altitudes.

### Step-by-Step Solution:

1. **Understanding the Orthocenter**: The orthocenter of a triangle is the point where the three altitudes intersect. In triangle ABC, the altitudes are AD, BE, and CF, with H being the orthocenter.

2. **Drawing Triangle ABC**: Start by sketching triangle ABC. Label the vertices A, B, and C.

3. **Constructing the Altitudes**:
- Draw altitude AD from vertex A to side BC.
- Draw altitude BE from vertex B to side AC.
- Draw altitude CF from vertex C to side AB.

4. **Identifying the Orthocenter H**: The point where all three altitudes intersect is the orthocenter H of triangle ABC.

5. **Finding the Orthocenter of Triangle ABH**:
- Now, we need to determine the orthocenter of triangle ABH.
- The vertices of triangle ABH are A, B, and H.

6. **Analyzing the Altitudes of Triangle ABH**:
- The altitude from A in triangle ABH will be perpendicular to line BH.
- The altitude from B will be perpendicular to line AH.
- The altitude from H will be perpendicular to line AB.

7. **Identifying C as the Orthocenter**:
- For C to be the orthocenter of triangle ABH, the following must hold:
- The line from C to AB must be perpendicular to AB.
- The line from C to A must be perpendicular to BH.
- The line from C to B must be perpendicular to AH.
- Since these conditions are satisfied, we conclude that C is indeed the orthocenter of triangle ABH.

### Conclusion:
Thus, the answer is that C is the orthocenter of triangle ABH.

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