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).
Altshiller-Court, N. College Geometry: A Second Course in Plane Geometry for Colleges
and Normal Schools, 2nd ed., rev. enl. New York: Barnes and Noble, pp. 93-97,
1952.
Brand, L. "The Eight-Point Circle and the Nine-Point Circle." Amer.
Math. Monthly 51, 84-85, 1944.
Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing
an Easy Introduction to Modern Geometry with Numerous Examples, 5th ed., rev. enl.
Dublin: Hodges, Figgis, & Co., pp. 58-61, 1888.
Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New
York: Chelsea, pp. 40-41, 1971.
Coxeter, H. S. M. and Greitzer, S. L. "The Nine-Point Circle." §1.8 in Geometry Revisited. New York: Random House, pp. 20-22,
1967.
Dörrie, H. "The Feuerbach Circle." §28 in 100 Great Problems of Elementary Mathematics: Their History and
Solutions. New York: Dover, pp. 142-144, 1965.
Durell, C. V. Modern Geometry: The Straight Line and Circle. London:
Macmillan, pp. 27-29, 1928.
F. Gabriel-Marie. Exercices de géométrie. Tours, France:
Maison Mame, pp. 306-314, 1912.
Gardner, M. Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles
from Scientific American. New York: Vintage Books, p. 59, 1977.
Guggenbuhl, L. "Karl Wilhelm Feuerbach, Mathematician." Appendix to Circles:
A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., pp. 89-100,
1995.
Honsberger, R. "The Nine-Point Circle." §1.3 in Episodes in Nineteenth and Twentieth Century Euclidean Geometry.
Washington, DC: Math. Assoc. Amer., pp. 6-7, 1995.
Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the
Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 165 and 195-212,
1929.
Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129,
1-295, 1998.
Lachlan, R. "The Nine-Point Circle." §123-125 in An Elementary Treatise on Modern Pure Geometry. London:
Macmillian, pp. 70-71, 1893.
Lange, J. Geschichte des Feuerbach'schen Kreises. Berlin, 1894.
Lemoine, M. T. "Note de géométrie." Nouv. Ann. Math. 4,
400-402, 1904.
Mackay, J. S. "History of the Nine-Point Circle." Proc. Edinburgh
Math. Soc. 11, 19-61, 1892.
Ogilvy, C. S. Excursions in Geometry. New York: Dover, pp. 119-120,
1990.
Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC:
Math. Assoc. Amer., pp. 1-4, 1995.
Rouché, E. and de Comberousse, C. Traité de géométrie plane. Paris:
Gauthier-Villars, pp. 306-307, 1900.
Schröder, E. M. "Zwei 8-Kreise-Sätze für Vierecke."
Mitt. Math. Ges. Hamburg 18, 105-117, 1999.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers.
Middlesex, England: Penguin Books, pp. 73-74, 1986.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry.
London: Penguin, pp. 158-159, 1991.
|
Contact the MathWorld Team
