Let H be the orthocentre of ΔABC. What is the orthocentre of triangle HBC?

Introduction / Context:Orthocentre transformations under triangle permutations are classic: if H is the orthocentre of ABC, then swapping in H with another vertex creates a triangle whose orthocentre is one of the original vertices.

Given Data / Assumptions:

• H is the orthocentre of ΔABC (intersection of altitudes).
• We consider ΔHBC.

Concept / Approach:Lines BH and CH are altitudes of ΔABC. In ΔHBC, the altitudes through B and C are along BA and CA respectively (since BH ⟂ AC and CH ⟂ AB). The third altitude of ΔHBC then passes through A, and the three altitudes of ΔHBC meet at A.

Step-by-Step Solution:In ΔHBC, altitude from B is along BA (perpendicular to HC/AB).Altitude from C is along CA.Their intersection is A; hence A is the orthocentre of ΔHBC.

Verification / Alternative check:Vector or coordinate proofs reflect the same concurrency.

Why Other Options Are Wrong:Points N, M, L are generic diagram labels without this universal property.

Common Pitfalls:Assuming H remains the orthocentre after vertex replacement; it does not in general.

Final Answer:A