Orthocentre of a Triangle

Learn what the orthocentre of a triangle is, how altitudes form it, and how its position changes with different triangle types, explained with simple diagrams and student-friendly notes.

1. What is the Orthocentre?

The orthocentre of a triangle is the point where the three altitudes of the triangle intersect. An altitude is a perpendicular drawn from a vertex to the opposite side (or the line containing that side).

        A
       /|
      / |\
     /  | \
    B---O---C
         O = orthocentre

2. Definition of Orthocentre

Definition: The orthocentre of a triangle is the common point of intersection of the three altitudes of the triangle.

If altitudes from vertices \(A\), \(B\), and \(C\) meet at point \(H\), then \(H\) is the orthocentre of \(\triangle ABC\).

3. Properties of the Orthocentre

  • It is the intersection point of the three altitudes.
  • The orthocentre’s location depends strongly on the type of triangle.
  • It can lie inside, on, or outside the triangle.
  • Altitudes always intersect at a single point (they are concurrent).

3.1. Altitude Behaviour

Each altitude is drawn from a vertex and meets the opposite side (or its extension) at a right angle (\(90^\circ\)). The orthocentre is the single point where all three such altitudes meet.

4. Location of the Orthocentre in Different Triangles

The orthocentre changes position depending on the type of triangle:

4.1. Acute-Angled Triangle

All angles are less than \(90^\circ\). The orthocentre lies inside the triangle.

4.2. Right-Angled Triangle

One angle is exactly \(90^\circ\). The orthocentre lies at the right-angle vertex. This happens because the two sides forming the right angle act as altitudes themselves.

4.3. Obtuse-Angled Triangle

One angle is greater than \(90^\circ\). The orthocentre lies outside the triangle. This is because one altitude has to be extended outside the boundary to meet the opposite side’s extension.

5. Constructing the Orthocentre

To construct the orthocentre of \(\triangle ABC\):

  1. Draw the altitude from vertex \(A\) to side \(BC\).
  2. Draw the altitude from vertex \(B\) to side \(AC\).
  3. The point where these two altitudes meet is the orthocentre \(H\).
  4. (Drawing the third altitude will also pass through the same point.)

6. Why the Orthocentre is Important

The orthocentre helps in:

  • studying triangle geometry and concurrency of lines,
  • understanding how altitudes behave in different triangles,
  • proofs involving right triangles,
  • geometric constructions where perpendicular heights are needed.

It is one of the four major centres of a triangle (others are centroid, incentre and circumcentre).