Trigonometry/For Enthusiasts/Erdős–Mordell inequality

In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point O inside ABC, the sum of the distances from O to the sides is less than or equal to half of the sum of the distances from O to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős 1935 posed the problem of proving the identity; a proof was provided two years later by Mordell D. F. Barrow 1937 txt. This solution was however not very elementary. Subsequent simpler proofs were then found by Kazarinoff 1957, Bankoff 1958, and Alsina Nelson 2007. In absolute geometry, the Erdős–Mordell inequality is equivalent to the statement that the sum of the angles of a triangle is at most 2 Pambuccian 2008. References Alsina Claudi Nelsen Roger B. Forum Geometricorum 99–102 A visual proof of the Erdős-Mordell inequality http://forumgeom.fau.edu/FG2007volume7/FG200711index.html 7 2007. Bankoff Leon Leon Bankoff American Mathematical Monthly 521 An elementary proof of the Erdős-Mordell theorem 7 http://www.jstor.org/stable/2308580 65 1958. Erdős Paul Paul Erdős American Mathematical Monthly 396 Problem 3740 42 1935. Kazarinoff D. K. 10.1307/mmj/1028988998 2 Michigan Mathematical Journal 97–98 A simple proof of the Erdős-Mordell inequality for triangles 4 1957. Mordell L. J. Louis Mordell Barrow D. F. American Mathematical Monthly 252–254 Solution to 3740 44 1937. Pambuccian Victor 10.1007/s00022-007-1961-4 Journal of Geometry 134–139 The Erdős-Mordell inequality is equivalent to non-positive curvature 88 2008.