HMMT 十一月 2011 · 团队赛 · 第 9 题

英文解析

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 .
   Answer:48
   B17
   A CMG
   ′
   ′G
   Let M be the midpoint of AC . For any triangle, we know that the centroid is located 2 / 3 of the way b
   ′ ′ ′ ′
   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, wecalculate 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.