The four notable points

Every triangle has four notable points, each obtained by intersecting a particular triple of lines.

In brief — The notable points

The four notable points of a triangle are:

• Centroid $G$: intersection of the medians. It divides each median in ratio $2:1$ starting from the vertex.
• Orthocentre $H$: intersection of the altitudes.
• Incentre $I$: intersection of the bisectors of the interior angles. It is the centre of the inscribed circle.
• Circumcentre $O$: intersection of the perpendicular bisectors of the sides. It is the centre of the circumscribed circle.

In a generic triangle the four notable points are distinct.

Remark — Equilateral triangle

In a generic triangle the four notable points are all distinct. In an equilateral triangle they all coincide at the same point.

In the equilateral triangle the four notable points coincide; the circumscribed and inscribed circles are also shown.

Links

Topics: Euclidean circle
Concepts: Centroid · Circumcentre · Incentre · Orthocentre · Notable points of a triangle
Skills: Synthetic geometry