HMMT 二月 2005 · 几何 · 第 3 题

英文解析

3. Let ABCD be a rectangle with area 1, and let E lie on side CD . What is the area ofthe triangle formed by the centroids of triangles ABE , BCE , and ADE ?
   Solution: 1/9
   Let the centroids of ABE , BCE , and ADE be denoted by X , Y , and Z , respectively.
   Let d ( P, QR ) denote the distance from P to line QR . Since the centroid lies two-thirdsof the distance from each vertex to the midpoint of the opposite edge, d ( X, AB ) =
   d ( Y, CD ) = d ( Z, CD ) = BC , so Y Z is parallel to CD and d ( X, Y Z ) = BC −1
   2 1 1 13
   BC = BC . Likewise, d ( Z, AD ) = DE and d ( Y, BC ) = CE , so that since Y Z
   3 3 3 3
   1 2
   is perpendicular to AD and BC , we have that Y Z = CD − ( DE + CE ) = CD .
   3 3
   1 1 2 1 1
   Therefore, the area of XY Z is ( BC )( CD ) = BC · CD = .
   2 3 3 9 9
   D ECBA
   ° °
   6 6