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埃爾德什-莫德爾定理


Erdos-MordellTheorem

如果 P 是三角形 DeltaABC 內的 垂足點,且 P_AP_BP_C 分別是從 P 到邊 BCCAAB 的垂線的垂足,則

 PA+PB+PC>=2(PP_A+PP_B+PP_C).

這個不等式由埃爾德什 (Erdős) 於 1935 年提出,並由莫德爾 (Mordell) 和巴羅 (Barrow) 於兩年後的 1937 年解決。隨後,卡扎裡諾夫 (Kazarinoff) 於 1945 年(Kazarinoff 1961, p. 78)和班科夫 (Bankoff) (1958) 找到了初等證明。

奧本海姆 (Oppenheim) (1961) 和莫德爾 (Mordell) (1962) 也證明了

 PA×PB×PC>=(PP_B+PP_C)(PP_C+PP_A)(PP_A+PP_B).

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參考文獻

Avez, A. "A Short Proof of a Theorem of Erdős and Mordell." Amer. Math. Monthly 100, 60-62, 1993.Bankoff, L. "An Elementary Proof of the Erdős-Mordell Theorem." Amer. Math. Monthly 65, 521, 1958.Brabant, H. "The Erdős-Mordell Inequality Again." Nieuw Tijdschr. Wisk. 46, 87, 1958/1959.Carlitz, L. "Some Inequalities for a Triangle." Amer. Math. Monthly 71, 881-885, 1964.Casey, J. A Sequel to the First Six Books of the Elements of Euclid, Containing an Easy Introduction to Modern Geometry with Numerous Examples, 6th ed. Dublin: Hodges, Figgis, & Co., p. 253, 1892.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 9, 1969.Dar, S. and Gueron, S. "A Weighted Erdős-Mordell Inequality." Amer. Math. Monthly 108, 165-167, 2001.Erdős, P. "Problem 3740." Amer. Math. Monthly 42, 396, 1935.Fejes Tóth, G. Lagerungen in der Ebene, auf der Kugel und in Raum, 2nd ed. Berlin: Springer-Verlag, 1972.Janous, W. "Further Inequalities of Erdos-Mordell Type." Forum Geom. 4, 203-206, 2004. http://forumgeom.fau.edu/FG2004volume4/FG200423index.html.Kazarinoff, D. K. "A Simple Proof of the Erdős-Mordell Inequality for Triangles." Michigan Math. J. 4, 97-98, 1957.Kazarinoff, N. D. Geometric Inequalities. New York: Random House, pp. 78-87, 1961.Komornik, V. "A Short Proof of the Erdős-Mordell Theorem." Amer. Math. Monthly 104, 57-60, 1997.Kontogiannis, D. G. Equalities and Inequalities in the Triangle. Athens: Ekpaideutikis, pp. 127-128, 1996.Lee, H. "Another Proof of the Erdős-Mordell Inequality." Forum Geom. 1, 7-8, 2001. http://forumgeom.fau.edu/FG2001volume1/FG200102index.html.Mordell, L. J. "On Geometric Problems of Erdős and Oppenheim." Math. Gaz. 46, 213-215, 1962.Mordell, L. J. and Barrow, D. F. "Solution to Problem 3740." Amer. Math. Monthly 44, 252-254, 1937.Oppenheim, A. "The Erdős Inequality and Other Inequalities for a Triangle." Amer. Math. Monthly 68, 226-230 and 349, 1961.Veldkamp, G. R. "The Erdős-Mordell Inequality." Nieuw Tijdschr. Wisk. 45, 193-196, 1957/1958.

請引用為

Weisstein, Eric W. "埃爾德什-莫德爾定理。" 來自 Web 資源。 https://mathworld.tw/Erdos-MordellTheorem.html

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