The Circle, also called Euler's Circle and the Feuerbach Circle, which passes through the feet of the Perpendicular , , and dropped from the Vertices of any 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 , , of the segments which join the Vertices and the Orthocenter . These three triples of points make nine in all, giving the 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's Hyperbola lies on the nine-point circle. The nine-point circle bisects any line from the Orthocenter to a point on the Circumcircle. The nine-point circle of the Incenter and Excenters of a Triangle is 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 Orthocenter have the same nine-point circle.

**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.

© 1996-9

1999-05-25