Nine-Point Circle
The Circle, also called Euler's Circle and the Feuerbach Circle, which passes through the feet of thePerpendicular 
, 
, and 
dropped from the Vertices ofany Triangle 
on the sides opposite them. Euler 
showed in 1765 that it also passes through theMidpoints 
, 
, 
of the sides of .
By Feuerbach's Theorem, the nine-pointcircle also passes through the Midpoints 
, 
, 
of the segments which join theVertices and the Orthocenter 
. These three triples of points make nine in all, givingthe circle its name. The center 
of the nine-point circle is called the Nine-Point Center.
The Radius of the nine-point circle is 
, where 
is the Circumradius. The center of Kiepert'sHyperbola lies on the nine-point circle. The nine-point circle bisects any line from the Orthocenter to a point onthe Circumcircle. The nine-point circle of the Incenter and Excenters of a Triangleis the Circumcircle.
The sum of the powers of the Vertices with regard to the nine-point circle is
Also,
where
is the Nine-Point Center, 
are the Vertices, is the Orthocenter,and is the Circumradius. All triangles inscribed in a given Circle and having the same Orthocenterhave the same nine-point circle.See also Complete Quadrilateral, Eight-Point Circle Theorem, Feuerbach's Theorem, FontenéTheorems, Griffiths' Theorem, Nine-Point Center, Nine-Point Conic, OrthocentricSystem
References
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. Coxeter, H. S. M. and Greitzer, S. L. 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. 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. 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. Lange, J. Geschichte des Feuerbach'schen Kreises. Berlin, 1894. 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.
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