通常被称为费尔巴哈定理的有两条定理。第一条定理指出，穿过从任意三角形的顶点向对边所作垂线的垂足的圆，也穿过这些边的中点，以及连接顶点到垂线交点的线段的中点。这样的圆被称为九点圆。
最常被称为费尔巴哈定理的命题指出，任意三角形的九点圆内切于内切圆，外切于三个旁切圆。这个定理最早由费尔巴哈 (Feuerbach) (1822) 发表。已经给出了许多证明 (Elder 1960)，其中最简单的是 M'Clelland (1891, p. 225) 和 Lachlan (1893, p. 74) 提出的证明。
参见
旁切圆,
费尔巴哈点,
费尔巴哈三角形,
哈特圆,
内切圆,
中点,
九点圆,
垂线,
切线
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参考文献
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. 107, 273, and 290, 1952.Baker, H. F. Appendix to Ch. 12 in An Introduction to Plane Geometry. Cambridge, England: Cambridge University Press, 1943.Coolidge, J. L. A Treatise on the Geometry of the Circle and Sphere. New York: Chelsea, p. 39, 1971.Coxeter, H. S. M. and Greitzer, S. L. Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 117-119, 1967.Dixon, R. Mathographics. New York: Dover, p. 59, 1991.Durell, C. V. Modern Geometry: The Straight Line and Circle. London: Macmillan, p. 117, 1928.Elder, A. E. "Feuerbach's Theorem: A New Proof." Amer. Math. Monthly 67, 905-906, 1960.F. Gabriel-Marie. Exercices de géométrie. Tours, France: Maison Mame, pp. 595-597, 1912.Feuerbach, K. W. Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks, und mehrerer durch die bestimmten Linien und Figuren. Nürnberg, Germany: Riegel und Wiesner, 1822.Gallatly, W. "Feuerbach's Theorem." §63 in The Modern Geometry of the Triangle, 2nd ed. London: Hodgson, p. 41, 1913.Kroll, W. "Elementarer Beweis des Satzes von Feuerbach." Praxis der Math. 40, 251-254, 1998.Lachlan, R. An Elementary Treatise on Modern Pure Geometry. London: Macmillan, 1893.M'Clelland, W. J. A Treatise on the Geometry of the Circle and Some Extensions to Conic Sections by the Method of Reciprocation, with Numerous Examples. London: Macmillan, 1891.Rouché, E. and de Comberousse, C. Traité de géométrie plane. Paris: Gauthier-Villars, pp. 307-309, 1900.Sawayama, Y. "Démonstration élémentaire du théorème de Feuerbach." L'enseign. math. 7, 479-482, 1905.Sawayama, Y. "8 nouvelles démonstrations d'un théorème relatif au cercle des 9 points." L'enseign. math. 13, 31-49, 1911.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. Middlesex, England: Penguin Books, pp. 76-77, 1991.在 Wolfram|Alpha 中被引用
费尔巴哈定理
引用为
Weisstein, Eric W. "费尔巴哈定理。" 来自 MathWorld--一个 Wolfram Web 资源。 https://mathworld.net.cn/FeuerbachsTheorem.html
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