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perimeter Inequality [p(ABC) p(PQR) >= (p(DEF))^2]

Source: IMO Shortlist 2005, problem G7, created by Hojoo Lee

July 2, 2006
geometryinequalitiescircumcircleIMO Shortlist

Problem Statement

In an acute triangle ABCABC, let DD, EE, FF be the feet of the perpendiculars from the points AA, BB, CC to the lines BCBC, CACA, ABAB, respectively, and let PP, QQ, RR be the feet of the perpendiculars from the points AA, BB, CC to the lines EFEF, FDFD, DEDE, respectively.
Prove that p(ABC)p(PQR)≥(p(DEF))2p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}, where p(T)p\left(T\right) denotes the perimeter of triangle TT .
Proposed by Hojoo Lee, Korea
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