Erdős–Mordell inequality

In Euclidean geometry, the Erdős–Mordell inequality states that for any triangle ABC and point P inside ABC, the sum of the distances from P to the sides is less than or equal to half of the sum of the distances from P to the vertices. It is named after Paul Erdős and Louis Mordell. Erdős (1935) posed the problem of proving the inequality; 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 $\pi$ (Pambuccian 2008).

Barrow's inequality is a strengthened version of the Erdős–Mordell inequality in which the distances from P to the sides are replaced by the distances from P to the points where the angle bisectors of ∠APB, ∠BPC, and ∠CPA cross the sides. Although the replaced distances are longer, their sum is still less than or equal to half the sum of the distances to the vertices.

Proof

Let the sides of ABC be a, b, c, also let PA=p, PB=q, PC=r, d(P;BC)=x, d(P;CA)=y, d(P;AB)=z. First, we prove that

$cr\geq ax+by$ .

This is equivalent to

$\frac{c(r+z)}2\geq \frac{ax+by+cz}2$ .

The RHS is the area of triangle ABC, but on the LHS, r+z is at least the height of the triangle, consequently, the LHS cannot be smaller than the RHS. Now reflect P on the angle bisector at C. We find that cr $\geq$ ay+bx for P's reflection. Similarly, bq $\geq$ az+cx and ap $\geq$ bz+cy. We solve these inequalities for r, q, and p:

$r\geq (a/c)y+(b/c)x$ ,

$q\geq (a/b)z+(c/b)x$ ,

$p\geq (b/a)z+(c/a)y$ .

Adding the three up, we get

$p+q+r\geq (\frac b c+\frac c b)x+(\frac a c+\frac c a)y+(\frac a b+\frac b a)z$ .

Since the sum of a positive number and its reciprocal is at least 2, we are finished. Equality holds only for the equilateral triangle, where P is its centroid.

References

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, JSTOR 2308580 .
Erdős, Paul (1935), "Problem 3740", American Mathematical Monthly 42: 396, doi:10.2307/2301373 .
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, doi:10.2307/2300713 .
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 .

External links

Weisstein, Eric W., "Erdős-Mordell Theorem", MathWorld.
Alexander Bogomolny, "Erdös-Mordell Inequality", from Cut-the-Knot.