Two new weighted Erdős-Mordell type inequalities (Q1745205)

The famous Erdős-Mordell inequality states that \(R_1+R_2+R_3\geq 2(r_1+r_2+r_3)\), where \(R_1,R_2,R_3\) are the distances of a point P, interior of a triangle ABC, from the vertices A, B, C and \(r_1,r_2,r_3\) are the distances from P to the sides BC, CA, AB. From the moment that the inequality was published in 1935 by Paul Erdős in the journal American Mathematical Monthly a large number of researcher was concerned with different proofs, various generalizations, and refinements of the inequality. The author of this article establishes two new Erdős-Mordell-type inequalities with the help of the software Maple. Two other interesting conjectures checked by computer are also proposed.