A New Sharpening of the Erdös-mordell Inequality and Related Inequalities

The famous Erdös-Mordell inequality in geometric inequalities states the following: Let P be an interior point of a triangle ABC. Let R1, R2, R3 be the distances from P to the vertices A,B,C, and let r1, r2, r3 be the distance from P to the sides BC,CA,AB, respectively. Then R1 +R2 +R3 ≥ 2(r1 + r2 + r3), (1.1) with equality if and only if △ABC is equilateral and P is its center. This equality was conjectured by Erdös [1] in 1935 and was first proved by Mordell [2] in the same year. Since then, a lot of proofs have been given, various generalizations, refinements and variations were also studied (see [3]-[29]). We recall here some related results. In [2], Barrow proved the stronger inequality: R1 +R2 +R3 ≥ 2(w1 + w2 + w3), (1.2) where w1, w2, w3 are the internal angle-bisectors of ∠BPC,∠CPA,∠APB respectively. The authors of the monograph [13] gave the following weighted generalization: x2R1 + y 2R2 + z 2R3 ≥ 2(yzr1 + zxr2 + xyr3), (1.3) with equality if and only if P is the circumcenter of △ABC and x : y : z = sinA : sinB : sinC, where A,B,C denote the angles of △ABC. In a recent paper [31], the author pointed out the following refinements: R1 +R2 +R3 ≥ 1 2 (