The nine point circle also known as the Euler's circle or the Feuerbach circle is the circle that passes through the feet of perpendiculars from the vertices and of a triangle

Some of the properties of this circle are:

Property 1 : This circle also passes through the midpoints of the sides and of This was shown by Euler.

Property 2 : Feuerbach showed that this circle also passes through the midpoints of the line segments and which are drawn from the vertices of to its orthocenter

These three triples of points make nine in all, giving the circle its name.

Property 3 : The radius of the nine-point cirlce is where is the circumradius (radius of the circumcircle).

Property 4 : The center of the nine-point circle is the midpoint of the line segment joining the orthocenter and the circumcenter, and hence lies on the Euler line.

Property 5 : All triangles inscribed in a given circle and having the same orthocenter, have the same nine-point circle.