PUMaC 2010 · 几何（B 组） · 第 8 题

PUMaC 2010 — Geometry (Division B) — Problem 8

Point P is in the interior of 4 ABC . The side lengths of ABC are AB = 7, BC = 8, CA = 9. The three foots of perpendicular lines from P to sides BC , CA , AB are D , E , F respectively. √ BC CA AB a Suppose the minimal value of + + can be written as c , where gcd( a, b ) = 1 and P D P E P F b c is square free, calculate abc . 2

解析

Point P is in the interior of 4 ABC . The side lengths of ABC are AB = 7, BC = 8, CA = 9. The three foots of perpendiculars from P to sides BC , CA , AB are D , E , F respectively. √ BC CA AB a Suppose the minimal value of + + can be written as c , where gcd( a, b ) = 1 and P D P E P F b c is square free, calculate abc . [Answer] 600 1 [Solution] Let S denote the area of 4 ABCR . Then S = ( BC · P D + CA · P E + AB · F P ). 2 By Cauchy-Schwarz inequality, ( ) BC CA AB 2 ( BC + CA + AB ) ≤ + + ( BC · P D + CA · P E + AB · F P ) , P D P E P F 3 2 BC CA AB ( BC + CA + AB ) hence, + + ≥ . So all we need to know is area of ABC , P D P E P F 2 S √ √ which by Heron’s formula is 12(12 − 7)(12 − 8)(12 − 9) = 12 5. Therefore, minimal value 2 √ (7 + 8 + 9) 24 is √ = 5. 5 24 5 4