Nine-Point Circle
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).
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nine-point circle
