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AP x BP x CP >= 8PL x PM x PN

Source: 2011 Saudi Arabia Pre-TST February 3.3 https://artofproblemsolving.com/community/c2745403_2011_

January 1, 2022
geometrygeometric inequality

Problem Statement

Let PP be a point in the interior of triangle ABCABC. Lines APAP, BPBP, CPCP intersect sides BCBC, CACA, ABAB at LL, MM, NN, respec­tively. Prove that AP⋅BP⋅CP≥8PL⋅PM⋅PN.AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.
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