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Nine-Point Circle


Nine-PointCircle

The nine-point circle, also called Euler's circle or the Feuerbach circle, is the circle that passes through the perpendicular feet H_A, H_B, and H_C dropped from the vertices of any reference triangle DeltaABC on the sides opposite them. Euler showed in 1765 that it also passes through the midpoints M_A, M_B, M_C of the sides of DeltaABC. By Feuerbach's theorem, the nine-point circle also passes through the midpoints E_A, E_B, and E_C of the segments that join the vertices and the orthocenter H. 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

 l=-1/2cosA,
(1)

giving the equation

 calphabeta+abetagamma+bgammaalpha-1/2(aalpha+bbeta+cgamma)(alphacosA+betacosB+gammacosC)=0.
(2)

The center N of the nine-point circle is called the nine-point center, and is Kimberling center X_5. The radius of the nine-point circle is

 R_N=1/2R,
(3)

where R is the circumradius of the reference triangle.

The midpoint of the two Fermat points X and X^' 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 X_i for i=11 (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 I is the incenter and J_A, J_B, and J_C are the excenters of a reference triangle DeltaABC, then the nine-point circles of triangles DeltaJ_AJ_BJ_C, DeltaIJ_BJ_C, DeltaIJ_CJ_A, and DeltaIJ_AJ_B all coincide with the circumcircle of DeltaABC.

FeuerbachTriangle

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).

NinePointCircles

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

 p_A+p_B+p_C=1/4(a^2+b^2+c^2).
(4)

Also,

 NA^2+NB^2+NC^2+NH^2=3R^2,
(5)

where N is the nine-point center, H is the orthocenter, and R 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

 alpha=1/(a(-a^4+b^2c^2+a^2c^2+a^2b^2))
(6)

(F. van Lamoen, pers. comm., Jan. 28, 2005), which is not a Kimberling center, but is the isotomic conjugate of X_(1078) and lies on lines (4, 160), (5, 141), (53, 232), (66, 2548), (184, 2980), (232, 427), and (311, 325).


See also

Apollonius Point, Complete Quadrilateral, Eight-Point Circle Theorem, Euler Points, Feuerbach's Theorem, Feuerbach Triangle, Fontené Theorems, Griffiths' Theorem, Hart Circle, Nine-Point Center, Nine-Point Conic, Orthocentric System, Rectangular Hyperbola

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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.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.

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Nine-Point Circle

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Weisstein, Eric W. "Nine-Point Circle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Nine-PointCircle.html

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