In a triangle, the orthocenter is the point where all three altitudes intersect. Now, let's consider the altitudes and the properties of the orthocenter in triangle ABC, with AD, BE, and CF as altitudes.

Given:

• $O$ is the orthocenter of triangle ABC.
• We need to find which triangle among the options has $B$ as its orthocenter.

Analysis:

1. Orthocenter properties:

   • In triangle ABC, $O$ is the orthocenter.
   • When we look at triangle BOC (with vertices at B, O, and C), its orthocenter will correspond to the original vertex $B$ of triangle ABC because the altitudes of triangle ABC pass through the orthocenter $O$.

2. Checking the options:

   • For triangle BOC, the altitude from $B$ in triangle ABC corresponds to an altitude in triangle BOC.
   • This gives us the orthocenter of triangle BOC as $B$.

Hence, the triangle that has the orthocenter at $B$ is:

Answer: (a) triangle BOC

Would you like further details or have any questions?

---

Here are 5 related questions to expand your understanding:

1. How do you locate the orthocenter in any triangle?
2. What are the special properties of the orthocenter in an acute triangle?
3. How do the orthocenter and centroid differ in terms of their location in a triangle?
4. Can the orthocenter of a triangle lie outside the triangle? If yes, under what conditions?
5. What is the relationship between the orthocenter and circumcenter of a triangle?

Tip: The orthocenter lies inside the triangle if it's acute, on the vertex of a right triangle, and outside for an obtuse triangle.