Incentre of a Triangle

The incentre of a triangle is the point where the three angle bisectors of the triangle meet. It is the centre of the triangle’s incircle — the largest circle that fits perfectly inside the triangle.

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

Definition: The incentre of a triangle is the point of intersection of the three internal angle bisectors.

If angle bisectors from vertices \(A\), \(B\), and \(C\) meet at \(I\), then \(I\) is the incentre of \(\triangle ABC\).

• The incentre always lies inside the triangle, irrespective of the type of triangle.
• It is the centre of the triangle’s incircle.
• The incentre is equidistant from all three sides of the triangle.
• A perpendicular from the incentre to any side gives the radius of the incircle.

The incircle is the largest circle that can be drawn inside a triangle such that it touches all three sides. The centre of this circle is the incentre.

The perpendicular radius of the incircle touches each side at exactly one point, called the point of tangency.

        A
       / \
      /   \
     /  o  \   (incircle)
    B-------C

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

1. Draw the angle bisector of \(\angle A\).
2. Draw the angle bisector of \(\angle B\).
3. Draw the angle bisector of \(\angle C\).
4. The point where they meet is the incentre.

Once the incentre is located, drop a perpendicular to any side to form the incircle’s radius.

The incentre helps in:

• constructing incircles of triangles,
• solving geometry problems involving distances to sides,
• understanding symmetry within a triangle,
• designing triangular spaces and boundaries in architecture.

Its equal-distance property makes it especially useful in design and construction.