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. posed the problem of proving the inequality; a proof was provided two years later by. This solution was however not very elementary. Subsequent simpler proofs were then found by, , and. 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.

Statement

Let P be an arbitrary point P inside a given triangle ABC, and let PL, PM, and PN be the perpendiculars from P to the sides of the triangles. (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

Proof

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. First, we prove that This is equivalent to The right side is the area of triangle ABC, but on the left side, r + z is at least the height of the triangle; consequently, the left side cannot be smaller than the right side. 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.

Another strengthened version

Let ABC be a triangle inscribed into a circle (O) and P be a point inside of ABC. 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

A generalization

Let be a convex polygon, and P be an interior point of. Let R_i be the distance from P to the vertex A_i, r_i the distance from P to the side A_iA_{i+1}, w_i the segment of the bisector of the angle A_iPA_{i+1} from P to its intersection with the side A_iA_{i+1} then :

In absolute geometry

In absolute geometry the Erdős–Mordell inequality is equivalent, as proved in, to the statement that the sum of the angles of a triangle is less than or equal to two right angles.