Erdös-Mordell Theorem

If $O$ is any point inside a Triangle $\Delta ABC$, and $P$, $Q$, and $R$ are the feet of the perpendiculars from $O$ upon the respective sides $BC$, $CA$, and $AB$, then

$$OA+OB+OC \geq 2(OP+OQ+OR).$$

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

$$OA\times OB\times OC\geq (OQ+OR)(OR+OP)(OP+OQ).$$

References

Bankoff, L. ``An Elementary Proof of the Erdös-Mordell Theorem.'' Amer. Math. Monthly 65, 521, 1958.

Brabant, H. ``The Erdös-Mordell Inequality Again.'' Nieuw Tijdschr. Wisk. 46, 87, 1958/1959.

Casey, J. A Sequel to the First Six Books of the Elements of Euclid, 6th ed. Dublin: Hodges, Figgis, & Co., p. 253, 1892.

Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 9, 1969.

Erdös, P. ``Problem 3740.'' Amer. Math. Monthly 42, 396, 1935.

Fejes-Tóth, L. Lagerungen in der Ebene auf der Kugel und im Raum. Berlin: Springer, 1953.

Mordell, L. J. ``On Geometric Problems of Erdös 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 Erdös Inequality and Other Inequalities for a Triangle.'' Amer. Math. Monthly 68, 226-230 and 349, 1961.

Veldkamp, G. R. ``The Erdös-Mordell Inequality.'' Nieuw Tijdschr. Wisk. 45, 193-196, 1957/1958.