HMMT 十一月 2011 · 团队赛 · 第 9 题

HMMT November 2011 — Team Round — Problem 9

[ 4 ] Let ABC be a triangle with AB = 9, BC = 10, and CA = 17. Let B be the reflection of the point ′ B over the line CA . Let G be the centroid of triangle ABC , and let G be the centroid of triangle ′ ′ AB C . Determine the length of segment GG .

解析

[ 4 ] Let ABC be a triangle with AB = 9, BC = 10, and CA = 17. Let B be the reflection of the point ′ B over the line CA . Let G be the centroid of triangle ABC , and let G be the centroid of triangle ′ ′ AB C . Determine the length of segment GG . 48 Answer: 17 B G M A C ′ G ′ B Let M be the midpoint of AC . For any triangle, we know that the centroid is located 2 / 3 of the way ′ ′ ′ ′ from the vertex, so we have M G/M B = M G /M B = 1 / 3, and it follows that M GG ∼ M BB . Thus, ′ ′ ′ GG = BB / 3 . However, note that BB is twice the altitude to AC in triangle ABC . To finish, we calculate the area of ABC in two different ways. By Heron’s Formula, we have √ [ ABC ] = 18(18 − 9)(18 − 10)(18 − 17) = 36, and we also have 1 17 ′ ′ [ ABC ] = BB · AC = ( BB ), 4 4 ′ ′ from which it follows that GG = BB / 3 = 48 / 17.