In a triangle, the orthocentre is defined as the point where all three altitudes meet. If the orthocentre of a triangle lies on one of its sides, what can you conclude about the position of the orthocentre in that triangle?

Introduction / Context:
This question tests knowledge of special points in a triangle, especially the orthocentre, which is the intersection of the three altitudes. The position of the orthocentre changes depending on whether the triangle is acute, right angled, or obtuse. Understanding these patterns is important for many geometry problems in competitive exams and helps build intuition about triangle centers.

Given Data / Assumptions:

• The orthocentre is the common intersection point of the three altitudes of a triangle.
• We are told that the orthocentre lies on one of the sides of the triangle.
• We work in standard Euclidean geometry.
• No additional special conditions about the triangle are provided.

Concept / Approach:
In an acute triangle, the orthocentre lies inside the triangle. In a right angled triangle, the orthocentre coincides with the right angled vertex. In an obtuse triangle, the orthocentre lies outside the triangle. The only time the orthocentre is said to lie on the boundary of the triangle, that is on a side, is when it coincides with a vertex, because every vertex lies on two sides. Thus, the condition that the orthocentre lies on a side strongly suggests that it is at one of the triangle vertices, as happens in a right angled triangle at the right angle vertex.

Step-by-Step Solution:
Step 1: Recall that in an acute triangle, all altitudes meet inside the triangle, so the orthocentre is interior. Step 2: Recall that in an obtuse triangle, the altitudes from acute angles intersect outside, so the orthocentre is exterior. Step 3: In a right angled triangle, the altitude from the right angle lies along one of the sides, and the other two altitudes intersect at the right angled vertex. Step 4: Therefore, in a right angled triangle, the orthocentre is located exactly at the right angled vertex. Step 5: A vertex lies on two sides of the triangle, so the statement that the orthocentre lies on one of the sides implies that it coincides with a vertex.

Verification / Alternative check:
We can sketch the three types of triangles. When we draw an acute triangle and its altitudes, the intersection point is clearly inside the triangle, not on a side. For an obtuse triangle, extending the altitudes shows the orthocentre outside the triangle. Only when we sketch a right angled triangle and draw all altitudes do we see that they meet at the right angle vertex, which lies on the two sides forming that right angle. This confirms that the given condition of the orthocentre lying on a side is satisfied when it coincides with a vertex.

Why Other Options Are Wrong:
The circumcentre necessarily lies outside the triangle: This happens only for obtuse triangles. In the current situation, the orthocentre is on a side, which is characteristic of a right angled triangle, not an obtuse one.
The circumcentre lies on the same side as the orthocentre: This is not a standard property and does not follow from the given condition.
The centroid always coincides with the orthocentre: The centroid is the intersection of medians and lies inside the triangle for all triangle types. It coincides with the orthocentre only in an equilateral triangle where the orthocentre is also inside, not on a side.
The incentre always coincides with the orthocentre: The incentre is the intersection of angle bisectors and is always inside the triangle. It does not coincide with the orthocentre in the general case implied by the question.

Common Pitfalls:
Students often mix up the properties of different triangle centers, such as centroid, circumcentre, incentre, and orthocentre. Another common error is to assume that lying on a side always means being in the middle of that side, whereas here it clearly refers to coincidence with a vertex. Carefully reviewing the positions of triangle centers in acute, right, and obtuse cases prevents such confusion.

Final Answer:
If the orthocentre lies on a side of the triangle, it must lie exactly at a vertex; thus the correct conclusion is the orthocentre lies exactly at a vertex of the triangle.