HMMT 二月 2003 · GEN2 赛 · 第 4 题

英文解析

4. In triangle ABC with area 51, points D and E trisect AB and points F and G trisect
   BC . Find the largest possible area of quadrilateral DEF G .
   Solution: 17
   Assume E is between D and B , and F is between G and B (the alternative is to switchtwo points, say D and E , which clearly gives a non-convex quadrilateral with smallerarea). If two triangles have their bases on the same line and the same opposite vertex,1
   then it follows from the bh formula that their areas are in the same ratio as their 1
   bases. In particular (brackets denote areas),2
   [ DBG ] [ DBG ] [ ABG ] DB BG 2 2 4 = · = · = · = ,
   [ ABC ] [ ABG ] [ ABC ] AB BC 3 3 9
   and similarly [ EBF ] / [ ABC ] = 1 / 9. Subtracting gives [ DEF G ] / [ ABC ] = 1 / 3, so theanswer is [ ABC ] / 3 = 17.