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Trigonometry/For Enthusiasts/Erdős–Mordell inequality

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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 and D. F. Barrow (1937). 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

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  • Alsina, Claudi; Nelsen, Roger B. (2007), "A visual proof of the Erdős-Mordell inequality", Forum Geometricorum, 7: 99–102.
  • Bankoff, Leon (1958), "An elementary proof of the Erdős-Mordell theorem", American Mathematical Monthly, 65 (7): 521.
  • Erdős, Paul (1935), "Problem 3740", American Mathematical Monthly, 42: 396.
  • Kazarinoff, D. K. (1957), "A simple proof of the Erdős-Mordell inequality for triangles", Michigan Mathematical Journal, 4 (2): 97–98, doi:10.1307/mmj/1028988998.
  • Mordell, L. J.; Barrow, D. F. (1937), "Solution to 3740", American Mathematical Monthly, 44: 252–254.
  • Pambuccian, Victor (2008), "The Erdős-Mordell inequality is equivalent to non-positive curvature", Journal of Geometry, 88: 134–139, doi:10.1007/s00022-007-1961-4.
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