In a triangle, the orthocentre is defined as the point where which set of line segments intersect?

Introduction / Context:
This conceptual question in geometry checks your understanding of special points associated with a triangle. In particular, it focuses on the orthocentre, which is one of the four main centres of a triangle, along with the centroid, circumcentre and incentre. Knowing the definitions of these points is crucial for solving many advanced geometry problems.

Given Data / Assumptions:

• We are considering an arbitrary triangle.
• Several important line segments can be drawn: medians, altitudes, perpendicular bisectors and angle bisectors.
• The question asks specifically about which set of segments intersect at the orthocentre.
• Standard triangle geometry in a Euclidean plane is assumed.

Concept / Approach:
Each notable triangle centre is associated with a specific type of segment:

• Centroid: intersection of the three medians.
• Orthocentre: intersection of the three altitudes.
• Circumcentre: intersection of the perpendicular bisectors of the sides.
• Incentre: intersection of the internal angle bisectors.

The orthocentre is the point where all three altitudes meet, where an altitude is a perpendicular drawn from a vertex to the opposite side (or its extension).

Step-by-Step Solution:
Step 1: Recall that an altitude of a triangle is a line drawn from a vertex perpendicular to the opposite side or its extended line. Step 2: Every triangle has three altitudes, one from each vertex. Step 3: In Euclidean geometry, these three altitudes are concurrent, meaning they intersect at a single point. Step 4: This common point of intersection of the three altitudes is defined as the orthocentre of the triangle. Step 5: Therefore, the orthocentre is the point where the altitudes meet.

Verification / Alternative check:
To visualize this, draw any acute triangle and construct its three altitudes using a right angle ruler. You will see that all three lines intersect at a single point inside the triangle. For a right angled triangle, this point coincides with the right angled vertex, and for an obtuse triangle, the orthocentre lies outside the triangle. In all cases, however, it is always the intersection of the altitudes.

Why Other Options Are Wrong:
The medians meet: Their intersection is the centroid, not the orthocentre.
The perpendicular bisectors of the sides meet: Their intersection point is the circumcentre.
The angle bisectors meet: Their intersection point is the incentre.
The internal and external bisectors meet: These describe various other points, not the orthocentre, and are not the standard definition of a single special centre like the orthocentre.

Common Pitfalls:
Students often confuse the different centres because each is defined by the concurrency of three lines. Mixing up medians and altitudes is especially common. A good way to remember is to associate each centre with its geometric role: centroid with balancing point, circumcentre with circumscribed circle, incentre with inscribed circle, and orthocentre with right angles (altitudes). Clear mental images and practice with diagrams can help keep these definitions straight.

Final Answer:
The orthocentre of a triangle is the point where the altitudes meet.