HMMT 二月 2003 · GEN2 赛 · 第 4 题

HMMT February 2003 — GEN2 Round — Problem 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 .

解析

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 switch two points, say D and E , which clearly gives a non-convex quadrilateral with smaller 1 area). 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 2 bases. In particular (brackets denote areas), [ 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 the answer is [ ABC ] / 3 = 17.