In geometry, we come across properties of triangles which sometimes include some special circles associated with a triangle, like the ‘circumcircle’. One such circle with some interesting properties is the nine-point circle.

The Nine Point Circle is a circle special to every triangle and passes through nine significant points:

1. The three midpoints of the sides of the triangles
2. The three-foot of the altitudes
3. The midpoint of the line joins the three vertices and the orthocenter.

(Here ‘P’ is the orthocenter, P1, P2, P3 are the foot of the perpendiculars, M1, M2, M3 are the midpoints of the sides, and MAP , MBP , MCP are the midpoint of the line joining the three vertices, A, B, C and the orthocenter respectively.)

Fig 1: The nine-point circle

An interesting property of the nine-point circle which relates the nine-point circle with the incircle and the excircles is Feuerbach’s Theorem.

Let’s first look into incircle and excircles.

An incircle is a circle inscribed in a triangle and an excircle is a circle touching the extended 2 sides of a triangle and the other side.  Feuerbach’s Theorem states ‘In any triangle, that the nine-point circle is tangent to the three excircles of the triangle as well as its incircle’ Fig 3: The Feuerbach’s Theorem – the figure shows only one excircle touching the other circles.