HMMT 二月 2015 · 几何 · 第 2 题

HMMT February 2015 — Geometry — Problem 2

Let ABC be a triangle with orthocenter H ; suppose that AB = 13, BC = 14, CA = 15. Let G be A the centroid of triangle HBC , and define G , G similarly. Determine the area of triangle G G G . B C A B C ◦

解析

Let ABC be a triangle with orthocenter H ; suppose that AB = 13, BC = 14, CA = 15. Let G be A the centroid of triangle HBC , and define G , G similarly. Determine the area of triangle G G G . 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 ] 1 so G G G is the dilation of ABC about H with ratio − . Thus [ G G G ] = . By Heron’s 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 ◦