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).

Subsequent simpler proofs were then found by Kazarinoff (1957), Bankoff (1958), and Alsina & Nelsen (2007).

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.

be an arbitrary point P inside a given triangle

(If the triangle is obtuse, one of these perpendiculars may cross through a different side of the triangle and end on the line supporting one of the sides.)

Then the inequality states that Let the sides of ABC be a opposite A, b opposite B, and c opposite C; also let PA = p, PB = q, PC = r, dist(P;BC) = x, dist(P;CA) = y, dist(P;AB) = z.

Now reflect P on the angle bisector at C. We find that cr ≥ ay + bx for P's reflection.

Similarly, bq ≥ az + cx and ap ≥ bz + cy.

We solve these inequalities for r, q, and p: Adding the three up, we get Since the sum of a positive number and its reciprocal is at least 2 by AM–GM inequality, we are finished.

Equality holds only for the equilateral triangle, where P is its centroid.

Let D, E, F be the orthogonal projections of P onto BC, CA, AB.

M, N, Q be the orthogonal projections of P onto tangents to (O) at A, B, C respectively, then: Equality hold if and only if triangle ABC is equilateral (Dao, Nguyen & Pham 2016; Marinescu & Monea 2017) Let

then (Lenhard 1961): In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in Pambuccian (2008), to the statement that the sum of the angles of a triangle is less than or equal to two right angles.