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p(A)p(B)p(C) /s^3 <= 1/8

Source: Indian Postal Coaching 2008 set 1 p2

May 25, 2020
geometryperpendiculardistanceGeometric Inequalities

Problem Statement

Let ABCABC be a triangle, ADAD be the altitude from AA on to BCBC. Draw perpendiculars DD1DD_1 and DD2DD_2 from DD on to ABAB and ACAC respectively and let p(A)p(A) be the length of the segment D1D2D_1D_2. Similarly define p(B)p(B) and p(C)p(C). Prove that p(A)p(B)p(C)s3≤18\frac{p(A)p(B)p(C)}{s^3}\le \frac18 , where s is the semi-perimeter of the triangle ABCABC.
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