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Erdős-Mordell Theorem

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Erdos-MordellTheorem

If P is a pedal point inside a triangle DeltaABC, and P_A, P_B, and P_C are the feet of the perpendiculars from P upon the respective sides BC, CA, and AB, then

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

This inequality was proposed by Erdős (1935), and solved by Mordell and Barrow (1937) two years later. Elementary proofs were subsequently found by Kazarinoff in 1945 (Kazarinoff 1961, p. 78) and Bankoff (1958).

Oppenheim (1961) and Mordell (1962) also showed that

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

See also

Pedal Point

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References

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.

Cite this as:

Weisstein, Eric W. "Erdős-Mordell Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Erdos-MordellTheorem.html

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