nine-point circle

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 $A, B$ and $C$ of a triangle $\triangle ABC.$

Some of the properties of this circle are:

Property 1 : This circle also passes through the midpoints of the sides $AB, BC$ and $CA$ of $\triangle ABC.$ This was shown by Euler.

Property 2 : Feuerbach showed that this circle also passes through the midpoints of the line segments $AH, BH$ and $CH$ which are drawn from the vertices of $\triangle ABC$ to its orthocenter $H.$

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

Property 3 : The radius of the nine-point cirlce is $R/2,$ where $R$ 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.