HMMT 二月 2007 · 冲刺赛 · 第 16 题

英文解析

16. [ 10 ] Let ABC be a triangle with AB = 7 , BC = 9 , and CA = 4. Let D be the point such that
    AB ‖ CD and CA ‖ BD . Let R be a point within triangle BCD . Lines and m going through R are ′ parallel to CA and AB respectively. Line meets AB and BC at P and P respectively, and m meets
    ′
    CA and BC at Q and Q respectively. If S denotes the largest possible sum of the areas of triangles
    ′ ′ ′ ′ 2
    BP P , RP Q , and CQQ , determine the value of S .
    ′ ′ ′4
    Answer: 180 . Let R denote the intersection of the lines through Q and P parallel to ` and m
    ′ ′ ′ ′ ′ ′ ′ ′ ′ ′
    respectively. Then [ RP Q ] = [ R P Q ]. Triangles BP P , R P Q , and CQQ lie in ABC withoutoverlap, so that on the one hand, S ≤ ABC . On the other, this bound is realizable by taking R to be avertex of triangle BCD . We compute the square of the area of ABC to be 10 · (10 − 9) · (10 − 7) · (10 − 4) =