The nine-point circle, also called Euler's circle or the Feuerbach circle, is the circle that passes through the perpendicular
feet 
,
, and 
dropped from the vertices of any reference
triangle 
on the sides opposite them. Euler showed in 1765 that it also passes through the
midpoints 
, 
, 
of the sides of 
. By Feuerbach's theorem,
the nine-point circle also passes through the midpoints 
, 
, and 
of the segments that join the vertices
and the orthocenter 
. These points are commonly referred to as the Euler
points.
These three triples of points make nine in all, giving the circle its name.
The nine-point circle is the complement of the circumcircle.
The nine-point circle has circle function
 |
(1)
|
giving the equation
 |
(2)
|
The center 
of the nine-point circle is called the nine-point
center, and is Kimberling center 
. The radius of the nine-point
circle is
 |
(3)
|
where 
is the circumradius of the reference
triangle.
The midpoint of the two Fermat points 
and 
lies on the nine-point circle, as does the intersection of the Euler
lines of the corner triangles determined by the vertices of a triangle and its
orthic triangle. The nine-point circle also passes
through Kimberling centers 
for 
(the Feuerbach point),
113, 114, 115 (center of the Kiepert hyperbola),
116, 117, 118, 119, 120, 121, 122, 123, 124, 125 (center of the Jerabek
hyperbola), 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138,
139, 1312, 1313, 1560, 1566, 2039, 2040, and 2679.
It is orthogonal to the Stevanović circle.
The nine-point circle bisects any line from the orthocenter to a point on the circumcircle.
If 
is the incenter and 
, 
, and 
are the excenters of a reference
triangle 
,
then the nine-point circles of triangles 
, 
, 
, and 
all coincide with the circumcircle
of 
.
The incircle and three excircles of a reference triangle are all touched by the nine-point circle. Furthermore, the three points on the nine-point circle that touch the excircles form the vertices of the Feuerbach triangle (Kimberling 1998, p. 158).
Given four arbitrary points, the four nine-points circles of the triangles formed by taking three points at a times are concurrent (Lemoine 1904; Wells 1991, p. 209; Schröder 1999). Moreover, if four points do not form an orthocentric system, then there is a unique rectangular hyperbola passing through them, and its center is given by the intersection of the nine-point circles of the points taken three at a time (Wells 1991, p. 209). Finally, the point of concurrence of the four nine-points circles is also the point of concurrence of the four circles determined by the feet of the perpendiculars dropped from each of the four points onto the sides of the triangle formed by the other three (Schröder 1999).
In a triangle, the sum of the circle powers of the vertices with regard to the nine-point circle is
 |
(4)
|
Also,
 |
(5)
|
where 
is the nine-point center, 
is the orthocenter, and 
is the circumradius.
All triangles inscribed in a given circle and having the same orthocenter have the same nine-point circle.
The perspector of the nine-point circle is the point with center function
 |
(6)
|
(F. van Lamoen, pers. comm., Jan. 28, 2005), which is not a Kimberling center, but is the isotomic conjugate of 
and lies on lines (4, 160), (5,
141), (53, 232), (66, 2548), (184, 2980), (232, 427), and (311, 325).
