HMMT 二月 2015 · 几何 · 第 2 题

英文解析

2. Let ABC be a triangle with orthocenter H ; suppose that AB = 13, BC = 14, CA = 15. Let G bethe centroid of triangle HBC , and define G , G similarly. Determine the area of triangle G G G .A
   B C A B C
   Answer: 28 / 3 Let D, E, F be the midpoints of BC , CA , and AB , respectively. Then G G G
   A B C
   2 1
   is the DEF about H with a ratio of , and DEF is the dilation of ABC about H with a ratio of − ,
   3 2
   [ ABC ]
   so G G G is the dilation of ABC about H with ratio − . Thus [ G G G ] = . By Heron’s 1
   A B C A B C
   3 9
   √
   formula, the area of ABC is 21 · 8 · 7 · 6 = 84, so the area of G G G is [ ABC ] / 9 = 84 / 9 = 28 / 3.
   A B C
   °